diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000..5b6a065
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1,4 @@
+.Rproj.user
+.Rhistory
+.RData
+.Ruserdata
diff --git a/hw-1.Rproj b/hw-1.Rproj
new file mode 100644
index 0000000..8e3c2eb
--- /dev/null
+++ b/hw-1.Rproj
@@ -0,0 +1,13 @@
+Version: 1.0
+
+RestoreWorkspace: Default
+SaveWorkspace: Default
+AlwaysSaveHistory: Default
+
+EnableCodeIndexing: Yes
+UseSpacesForTab: Yes
+NumSpacesForTab: 2
+Encoding: UTF-8
+
+RnwWeave: Sweave
+LaTeX: pdfLaTeX
diff --git a/index.html b/index.html
index 3a108ac..11031b2 100644
--- a/index.html
+++ b/index.html
@@ -2,25 +2,20 @@
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
 
 <meta charset="utf-8">
-<meta name="generator" content="quarto-1.2.313">
+<meta name="generator" content="quarto-1.0.36">
 
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
-<meta name="author" content="Insert your name here">
+<meta name="author" content="Avi Singh">
 
 <title>Homework 1</title>
 <style>
 code{white-space: pre-wrap;}
 span.smallcaps{font-variant: small-caps;}
-div.columns{display: flex; gap: min(4vw, 1.5em);}
-div.column{flex: auto; overflow-x: auto;}
+span.underline{text-decoration: underline;}
+div.column{display: inline-block; vertical-align: top; width: 50%;}
 div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
 ul.task-list{list-style: none;}
-ul.task-list li input[type="checkbox"] {
-  width: 0.8em;
-  margin: 0 0.8em 0.2em -1.6em;
-  vertical-align: middle;
-}
 pre > code.sourceCode { white-space: pre; position: relative; }
 pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
 pre > code.sourceCode > span:empty { height: 1.2em; }
@@ -59,7 +54,7 @@
 code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
 code span.at { color: #7d9029; } /* Attribute */
 code span.bn { color: #40a070; } /* BaseN */
-code span.bu { color: #008000; } /* BuiltIn */
+code span.bu { } /* BuiltIn */
 code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
 code span.ch { color: #4070a0; } /* Char */
 code span.cn { color: #880000; } /* Constant */
@@ -72,7 +67,7 @@
 code span.ex { } /* Extension */
 code span.fl { color: #40a070; } /* Float */
 code span.fu { color: #06287e; } /* Function */
-code span.im { color: #008000; font-weight: bold; } /* Import */
+code span.im { } /* Import */
 code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
 code span.kw { color: #007020; font-weight: bold; } /* Keyword */
 code span.op { color: #666666; } /* Operator */
@@ -117,18 +112,17 @@ <h1 class="title">Homework 1</h1>
       <div>
       <div class="quarto-title-meta-heading">Author</div>
       <div class="quarto-title-meta-contents">
-               <p><span style="background-color: yellow;">Insert your name here</span> </p>
+               <p><span style="background-color: yellow;">Avi Singh</span> </p>
             </div>
     </div>
       
-    
       
     </div>
     
   
   </header><div id="quarto-content" class="page-columns page-rows-contents page-layout-article">
 <div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
-  <nav id="TOC" role="doc-toc" class="toc-active">
+  <nav id="TOC" role="doc-toc">
     <h2 id="toc-title">Table of contents</h2>
    
   <ul>
@@ -202,7 +196,7 @@ <h2 class="anchored" data-anchor-id="question-1">Question 1</h2>
 <li><p>In order to activate the <code>R</code> environment for the homework, make sure you have <code>renv</code> installed beforehand. To activate the <code>renv</code> environment for this assignment, open an instance of the <code>R</code> console from within the directory and type</p>
 <div class="sourceCode" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>renv<span class="sc">::</span><span class="fu">activate</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <p>Follow the instrutions in order to make sure that <code>renv</code> is configured correctly.</p></li>
-<li><p>Work on the <em>reminaing part</em> of this assignment as a <code>.qmd</code> file.</p>
+<li><p>Work on the <em>remaining part</em> of this assignment as a <code>.qmd</code> file.</p>
 <ul>
 <li>Create a <code>PDF</code> and <code>HTML</code> file for your output by modifying the YAML frontmatter for the Quarto <code>.qmd</code> document</li>
 </ul></li>
@@ -256,6 +250,30 @@ <h2 class="anchored" data-anchor-id="question-2">Question 2</h2>
 <p>How many elements of <code>my_vec_double</code> are greater than zero?</p></li>
 <li><p>Sort the values of <code>my_vec_double</code> in ascending order.</p></li>
 </ol>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">typeof</span>(my_vec) <span class="co"># 1. The data type is character</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>[1] "character"</code></pre>
+</div>
+<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>my_vec_double <span class="ot">&lt;-</span> <span class="fu">as.double</span>(my_vec) <span class="co"># 2. Converting data types</span></span>
+<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>my_vec_int <span class="ot">&lt;-</span> <span class="fu">as.integer</span>(my_vec)</span>
+<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a><span class="fu">typeof</span>(my_vec_double) <span class="co"># Verifying the types of vectors are what we want</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>[1] "double"</code></pre>
+</div>
+<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="fu">typeof</span>(my_vec_int)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>[1] "integer"</code></pre>
+</div>
+<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>my_vec_bool <span class="ot">&lt;-</span> <span class="fu">ifelse</span>(my_vec_double <span class="sc">&gt;=</span> <span class="dv">0</span>, <span class="cn">FALSE</span>, <span class="cn">TRUE</span>)</span>
+<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a><span class="fu">summary</span>(my_vec_bool)[<span class="st">"FALSE"</span>] <span class="co"># 3. Count number of False occurrences (4)</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>FALSE 
+    4 </code></pre>
+</div>
+<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>my_vec_double_sorted <span class="ot">=</span> <span class="fu">sort</span>(my_vec_double) <span class="co"># 4. Sort values of my_vec_double</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
 <p><br><br><br><br> <br><br><br><br></p>
 <hr>
 </section>
@@ -276,7 +294,9 @@ <h2 class="anchored" data-anchor-id="question-3">Question 3</h2>
 </div>
 <p>In this question we will get a better understanding of how <code>R</code> handles large data structures in memory.</p>
 <ol type="1">
-<li>Provide <code>R</code> code to construct the following matrices: <span class="math display">\[
+<li>Provide <code>R</code> code to construct the following matrices:</li>
+</ol>
+<p><span class="math display">\[
 \begin{bmatrix}
 1 &amp; 2 &amp; 3\\
 4 &amp; 5 &amp; 6\\
@@ -287,8 +307,14 @@ <h2 class="anchored" data-anchor-id="question-3">Question 3</h2>
 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 &amp; \dots &amp; 100\\
 1 &amp; 4 &amp; 9 &amp; 16 &amp; 25 &amp; \dots &amp; 10000\\
 \end{bmatrix}
-\]</span></li>
-</ol>
+\]</span></p>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>mat1 <span class="ot">&lt;-</span> <span class="fu">matrix</span>(<span class="fu">c</span>(<span class="dv">1</span>,<span class="dv">2</span>,<span class="dv">3</span>,<span class="dv">4</span>,<span class="dv">5</span>,<span class="dv">6</span>,<span class="dv">7</span>,<span class="dv">8</span>,<span class="dv">9</span>),<span class="at">nrow=</span><span class="dv">3</span>, <span class="at">ncol=</span><span class="dv">3</span>, <span class="at">byrow =</span> <span class="cn">TRUE</span>) <span class="co"># Create first matrix</span></span>
+<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>mat2row1 <span class="ot">&lt;-</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">100</span></span>
+<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a>mat2row2 <span class="ot">&lt;-</span> (mat2row1)<span class="sc">^</span><span class="dv">2</span></span>
+<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a>mat2data <span class="ot">&lt;-</span> <span class="fu">c</span>(mat2row1, mat2row2)</span>
+<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a>mat2 <span class="ot">&lt;-</span> <span class="fu">matrix</span>(mat2data, <span class="at">nrow =</span> <span class="dv">2</span>, <span class="at">ncol =</span> <span class="dv">100</span>, <span class="at">byrow =</span> <span class="cn">TRUE</span>) <span class="co"># Create second matrix</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
 <div class="callout-warning callout callout-style-default callout-captioned">
 <div class="callout-header d-flex align-content-center">
 <div class="callout-icon-container">
@@ -304,91 +330,166 @@ <h2 class="anchored" data-anchor-id="question-3">Question 3</h2>
 </div>
 <p>In the next part, we will discover how knowledge of the way in which a matrix is stored in memory can inform better code choices. To this end, the following function takes an input <span class="math inline">\(n\)</span> and creates an <span class="math inline">\(n \times n\)</span> matrix with random entries.</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>generate_matrix <span class="ot">&lt;-</span> <span class="cf">function</span>(n){</span>
-<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span>(</span>
-<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>        <span class="fu">matrix</span>(</span>
-<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a>            <span class="fu">rnorm</span>(n<span class="sc">^</span><span class="dv">2</span>),</span>
-<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a>            <span class="at">nrow=</span>n</span>
-<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a>        )</span>
-<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a>    )</span>
-<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>generate_matrix <span class="ot">&lt;-</span> <span class="cf">function</span>(n){</span>
+<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span>(</span>
+<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a>        <span class="fu">matrix</span>(</span>
+<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a>            <span class="fu">rnorm</span>(n<span class="sc">^</span><span class="dv">2</span>),</span>
+<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a>            <span class="at">nrow=</span>n</span>
+<span id="cb14-6"><a href="#cb14-6" aria-hidden="true" tabindex="-1"></a>        )</span>
+<span id="cb14-7"><a href="#cb14-7" aria-hidden="true" tabindex="-1"></a>    )</span>
+<span id="cb14-8"><a href="#cb14-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
 <p>For example:</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fu">generate_matrix</span>(<span class="dv">4</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">generate_matrix</span>(<span class="dv">4</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>           [,1]       [,2]       [,3]        [,4]
-[1,]  0.6398861 0.35207981 -0.6863098 -0.01839937
-[2,]  0.7613301 0.72302916  1.5097731  0.48976233
-[3,] -0.2608060 0.53979026 -1.0638331 -0.42020192
-[4,]  0.6602682 0.08892854  2.3460769  0.03895466</code></pre>
+[1,]  1.4559777  3.2144540 -2.0936790 -1.20219448
+[2,] -0.5214841 -1.7977769  1.3761673 -0.01667557
+[3,]  1.1293354 -0.1978237 -0.2550914 -1.06344894
+[4,]  0.4088185  1.7878860  0.2596922 -1.04892109</code></pre>
 </div>
 </div>
 <p>Let <code>M</code> be a fixed <span class="math inline">\(50 \times 50\)</span> matrix</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>M <span class="ot">&lt;-</span> <span class="fu">generate_matrix</span>(<span class="dv">50</span>)</span>
-<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="fu">mean</span>(M)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>M <span class="ot">&lt;-</span> <span class="fu">generate_matrix</span>(<span class="dv">50</span>)</span>
+<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a><span class="fu">mean</span>(M)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code>[1] 0.03185503</code></pre>
+<pre><code>[1] 0.04899604</code></pre>
 </div>
 </div>
 <ol start="2" type="1">
 <li>Write a function <code>row_wise_scan</code> which scans the entries of <code>M</code> one row after another and outputs the number of elements whose value is <span class="math inline">\(\ge 0\)</span>. You can use the following <strong>starter code</strong></li>
 </ol>
-<div class="sourceCode" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>row_wise_scan <span class="ot">&lt;-</span> <span class="cf">function</span>(x){</span>
-<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>    n <span class="ot">&lt;-</span> <span class="fu">nrow</span>(x)</span>
-<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>    m <span class="ot">&lt;-</span> <span class="fu">ncol</span>(x)</span>
-<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a>    <span class="co"># Insert your code here</span></span>
-<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a>    count <span class="ot">&lt;-</span> <span class="dv">0</span></span>
-<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a>    <span class="cf">for</span>(...){</span>
-<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a>        <span class="cf">for</span>(...){</span>
-<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a>            <span class="cf">if</span>(...){</span>
-<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a>                count <span class="ot">&lt;-</span> count <span class="sc">+</span> <span class="dv">1</span> </span>
-<span id="cb9-11"><a href="#cb9-11" aria-hidden="true" tabindex="-1"></a>            }</span>
-<span id="cb9-12"><a href="#cb9-12" aria-hidden="true" tabindex="-1"></a>        }</span>
-<span id="cb9-13"><a href="#cb9-13" aria-hidden="true" tabindex="-1"></a>    }</span>
-<span id="cb9-14"><a href="#cb9-14" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb9-15"><a href="#cb9-15" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span>(count)</span>
-<span id="cb9-16"><a href="#cb9-16" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a>row_wise_scan <span class="ot">&lt;-</span> <span class="cf">function</span>(x){</span>
+<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a>    n <span class="ot">&lt;-</span> <span class="fu">nrow</span>(x)</span>
+<span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a>    m <span class="ot">&lt;-</span> <span class="fu">ncol</span>(x)</span>
+<span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb19-5"><a href="#cb19-5" aria-hidden="true" tabindex="-1"></a>    <span class="co"># Insert your code here</span></span>
+<span id="cb19-6"><a href="#cb19-6" aria-hidden="true" tabindex="-1"></a>    count <span class="ot">&lt;-</span> <span class="dv">0</span></span>
+<span id="cb19-7"><a href="#cb19-7" aria-hidden="true" tabindex="-1"></a>    <span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>n){</span>
+<span id="cb19-8"><a href="#cb19-8" aria-hidden="true" tabindex="-1"></a>        <span class="cf">for</span>(value <span class="cf">in</span> x[i,]){</span>
+<span id="cb19-9"><a href="#cb19-9" aria-hidden="true" tabindex="-1"></a>            <span class="cf">if</span>(value<span class="sc">&gt;=</span><span class="dv">0</span>){</span>
+<span id="cb19-10"><a href="#cb19-10" aria-hidden="true" tabindex="-1"></a>                count <span class="ot">&lt;-</span> count <span class="sc">+</span> <span class="dv">1</span> </span>
+<span id="cb19-11"><a href="#cb19-11" aria-hidden="true" tabindex="-1"></a>            }</span>
+<span id="cb19-12"><a href="#cb19-12" aria-hidden="true" tabindex="-1"></a>        }</span>
+<span id="cb19-13"><a href="#cb19-13" aria-hidden="true" tabindex="-1"></a>    }</span>
+<span id="cb19-14"><a href="#cb19-14" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb19-15"><a href="#cb19-15" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span>(count)</span>
+<span id="cb19-16"><a href="#cb19-16" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
 <ol start="3" type="1">
 <li>Similarly, write a function <code>col_wise_scan</code> which does exactly the same thing but scans the entries of <code>M</code> one column after another</li>
 </ol>
-<div class="sourceCode" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>col_wise_scan <span class="ot">&lt;-</span> <span class="cf">function</span>(x){</span>
-<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a>    count <span class="ot">&lt;-</span> <span class="dv">0</span></span>
-<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a>    </span>
-<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a>    ... <span class="co"># Insert your code here</span></span>
-<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span>(count)</span>
-<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a>col_wise_scan <span class="ot">&lt;-</span> <span class="cf">function</span>(x){</span>
+<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a>    count <span class="ot">&lt;-</span> <span class="dv">0</span></span>
+<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a>    m <span class="ot">&lt;-</span> <span class="fu">ncol</span>(x)</span>
+<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a>    <span class="cf">for</span>(i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>m){</span>
+<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a>      <span class="cf">for</span>(value <span class="cf">in</span> x[,i]){</span>
+<span id="cb20-6"><a href="#cb20-6" aria-hidden="true" tabindex="-1"></a>        <span class="cf">if</span>(value<span class="sc">&gt;=</span><span class="dv">0</span>){</span>
+<span id="cb20-7"><a href="#cb20-7" aria-hidden="true" tabindex="-1"></a>          count <span class="ot">&lt;-</span> count <span class="sc">+</span> <span class="dv">1</span></span>
+<span id="cb20-8"><a href="#cb20-8" aria-hidden="true" tabindex="-1"></a>        }</span>
+<span id="cb20-9"><a href="#cb20-9" aria-hidden="true" tabindex="-1"></a>      }</span>
+<span id="cb20-10"><a href="#cb20-10" aria-hidden="true" tabindex="-1"></a>    }</span>
+<span id="cb20-11"><a href="#cb20-11" aria-hidden="true" tabindex="-1"></a></span>
+<span id="cb20-12"><a href="#cb20-12" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span>(count)</span>
+<span id="cb20-13"><a href="#cb20-13" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
 <p>You can check if your code is doing what it’s supposed to using the function here<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a></p>
 <ol start="4" type="1">
-<li><p>Between <code>col_wise_scan</code> and <code>row_wise_scan</code>, which function do you expect to take shorter to run? Why?</p></li>
-<li><p>Write a function <code>time_scan</code> which takes in a method <code>f</code> and a matrix <code>M</code> and outputs the amount of time taken to run <code>f(M)</code></p></li>
+<li>Between <code>col_wise_scan</code> and <code>row_wise_scan</code>, which function do you expect to take shorter to run? Why?</li>
 </ol>
-<div class="sourceCode" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>time_scan <span class="ot">&lt;-</span> <span class="cf">function</span>(f, M){</span>
-<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>    initial_time <span class="ot">&lt;-</span> ... <span class="co"># Write your code here</span></span>
-<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">f</span>(M)</span>
-<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a>    final_time <span class="ot">&lt;-</span> ...  <span class="co"># Write your code here</span></span>
-<span id="cb12-5"><a href="#cb12-5" aria-hidden="true" tabindex="-1"></a>    </span>
-<span id="cb12-6"><a href="#cb12-6" aria-hidden="true" tabindex="-1"></a>    total_time_taken <span class="ot">&lt;-</span> final_time <span class="sc">-</span> initial_time</span>
-<span id="cb12-7"><a href="#cb12-7" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span>(total_time_taken)</span>
-<span id="cb12-8"><a href="#cb12-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<ul>
+<li>I would expect row_wise_scan to take shorter to run since R handles row-wise operations faster than column-wise.</li>
+</ul>
+<ol start="5" type="1">
+<li>Write a function <code>time_scan</code> which takes in a method <code>f</code> and a matrix <code>M</code> and outputs the amount of time taken to run <code>f(M)</code></li>
+</ol>
+<div class="sourceCode" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a>time_scan <span class="ot">&lt;-</span> <span class="cf">function</span>(f, M){</span>
+<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a>    initial_time <span class="ot">&lt;-</span> ... <span class="co"># Write your code here</span></span>
+<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">f</span>(M)</span>
+<span id="cb21-4"><a href="#cb21-4" aria-hidden="true" tabindex="-1"></a>    final_time <span class="ot">&lt;-</span> ...  <span class="co"># Write your code here</span></span>
+<span id="cb21-5"><a href="#cb21-5" aria-hidden="true" tabindex="-1"></a>    </span>
+<span id="cb21-6"><a href="#cb21-6" aria-hidden="true" tabindex="-1"></a>    total_time_taken <span class="ot">&lt;-</span> final_time <span class="sc">-</span> initial_time</span>
+<span id="cb21-7"><a href="#cb21-7" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span>(total_time_taken)</span>
+<span id="cb21-8"><a href="#cb21-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a>time_scan <span class="ot">&lt;-</span> <span class="cf">function</span>(f, M){</span>
+<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a>    initial_time <span class="ot">&lt;-</span> <span class="fu">Sys.time</span>()</span>
+<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">f</span>(M)</span>
+<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a>    final_time <span class="ot">&lt;-</span> <span class="fu">Sys.time</span>()</span>
+<span id="cb22-5"><a href="#cb22-5" aria-hidden="true" tabindex="-1"></a>    </span>
+<span id="cb22-6"><a href="#cb22-6" aria-hidden="true" tabindex="-1"></a>    total_time_taken <span class="ot">&lt;-</span> final_time <span class="sc">-</span> initial_time</span>
+<span id="cb22-7"><a href="#cb22-7" aria-hidden="true" tabindex="-1"></a>    <span class="fu">return</span>(total_time_taken)</span>
+<span id="cb22-8"><a href="#cb22-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+</div>
 <p>Provide your output to</p>
-<div class="sourceCode" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="fu">list</span>(</span>
-<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a>    <span class="at">row_wise_time =</span> <span class="fu">time_scan</span>(row_wise_scan, M),</span>
-<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a>    <span class="at">col_wise_time =</span> <span class="fu">time_scan</span>(row_wise_scan, M)</span>
-<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="fu">list</span>(</span>
+<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a>    <span class="at">row_wise_time =</span> <span class="fu">time_scan</span>(row_wise_scan, M),</span>
+<span id="cb23-3"><a href="#cb23-3" aria-hidden="true" tabindex="-1"></a>    <span class="at">col_wise_time =</span> <span class="fu">time_scan</span>(col_wise_scan, M)</span>
+<span id="cb23-4"><a href="#cb23-4" aria-hidden="true" tabindex="-1"></a>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>$row_wise_time
+Time difference of 0.003378153 secs
+
+$col_wise_time
+Time difference of 0.002463818 secs</code></pre>
+</div>
+</div>
 <p>Which took longer to run?</p>
+<p>row_wise_time took longer to run</p>
 <ol start="6" type="1">
 <li>Repeat this experiment now when:
 <ul>
 <li><code>M</code> is a <span class="math inline">\(100 \times 100\)</span> matrix</li>
 <li><code>M</code> is a <span class="math inline">\(1000 \times 1000\)</span> matrix</li>
-<li><code>M</code> is a <span class="math inline">\(5000 \times 5000\)</span> matrix</li>
+<li><code>M</code> is a <span class="math inline">\(5000 \times 5000\)</span> matrix ::: {.cell}</li>
 </ul></li>
 </ol>
+<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a>M_100 <span class="ot">&lt;-</span> <span class="fu">generate_matrix</span>(<span class="dv">100</span>)</span>
+<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a>M_1000 <span class="ot">&lt;-</span> <span class="fu">generate_matrix</span>(<span class="dv">1000</span>)</span>
+<span id="cb25-3"><a href="#cb25-3" aria-hidden="true" tabindex="-1"></a>M_5000 <span class="ot">&lt;-</span> <span class="fu">generate_matrix</span>(<span class="dv">5000</span>)</span>
+<span id="cb25-4"><a href="#cb25-4" aria-hidden="true" tabindex="-1"></a><span class="fu">list</span>(</span>
+<span id="cb25-5"><a href="#cb25-5" aria-hidden="true" tabindex="-1"></a>    <span class="at">row_wise_time =</span> <span class="fu">time_scan</span>(row_wise_scan, M_100),</span>
+<span id="cb25-6"><a href="#cb25-6" aria-hidden="true" tabindex="-1"></a>    <span class="at">col_wise_time =</span> <span class="fu">time_scan</span>(col_wise_scan, M_100)</span>
+<span id="cb25-7"><a href="#cb25-7" aria-hidden="true" tabindex="-1"></a>    )</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>$row_wise_time
+Time difference of 0.0002739429 secs
+
+$col_wise_time
+Time difference of 0.0002388954 secs</code></pre>
+</div>
+<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="fu">list</span>(</span>
+<span id="cb27-2"><a href="#cb27-2" aria-hidden="true" tabindex="-1"></a>    <span class="at">row_wise_time =</span> <span class="fu">time_scan</span>(row_wise_scan, M_1000),</span>
+<span id="cb27-3"><a href="#cb27-3" aria-hidden="true" tabindex="-1"></a>    <span class="at">col_wise_time =</span> <span class="fu">time_scan</span>(col_wise_scan, M_1000)</span>
+<span id="cb27-4"><a href="#cb27-4" aria-hidden="true" tabindex="-1"></a>    )</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>$row_wise_time
+Time difference of 0.02448511 secs
+
+$col_wise_time
+Time difference of 0.02439094 secs</code></pre>
+</div>
+<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="fu">list</span>(</span>
+<span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a>    <span class="at">row_wise_time =</span> <span class="fu">time_scan</span>(row_wise_scan, M_5000),</span>
+<span id="cb29-3"><a href="#cb29-3" aria-hidden="true" tabindex="-1"></a>    <span class="at">col_wise_time =</span> <span class="fu">time_scan</span>(col_wise_scan, M_5000)</span>
+<span id="cb29-4"><a href="#cb29-4" aria-hidden="true" tabindex="-1"></a>    )</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>$row_wise_time
+Time difference of 0.8133478 secs
+
+$col_wise_time
+Time difference of 0.6909711 secs</code></pre>
+</div>
+<p>:::</p>
 <p>What can you conclude?</p>
+<ul>
+<li>Generally speaking, row_wise_time takes longer to run, and the amount of time it takes to run increases with dimensions of M</li>
+</ul>
 <p><br><br><br><br> <br><br><br><br></p>
 <hr>
 </section>
@@ -396,18 +497,20 @@ <h2 class="anchored" data-anchor-id="question-3">Question 3</h2>
 <h1>Appendix</h1>
 <p>Print your <code>R</code> session information using the following command</p>
 <div class="cell">
-<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sessionInfo</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sessionInfo</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code>R version 4.2.2 (2022-10-31)
-Platform: x86_64-apple-darwin22.1.0 (64-bit)
-Running under: macOS Ventura 13.1
+<pre><code>R version 4.2.2 (2022-10-31 ucrt)
+Platform: x86_64-w64-mingw32/x64 (64-bit)
+Running under: Windows 10 x64 (build 22000)
 
 Matrix products: default
-BLAS:   /usr/local/Cellar/openblas/0.3.21/lib/libopenblasp-r0.3.21.dylib
-LAPACK: /usr/local/Cellar/r/4.2.2_1/lib/R/lib/libRlapack.dylib
 
 locale:
-[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
+[1] LC_COLLATE=English_United States.utf8 
+[2] LC_CTYPE=English_United States.utf8   
+[3] LC_MONETARY=English_United States.utf8
+[4] LC_NUMERIC=C                          
+[5] LC_TIME=English_United States.utf8    
 
 attached base packages:
 [1] stats     graphics  grDevices datasets  utils     methods   base     
@@ -415,23 +518,39 @@ <h1>Appendix</h1>
 loaded via a namespace (and not attached):
  [1] digest_0.6.31   lifecycle_1.0.3 jsonlite_1.8.4  magrittr_2.0.3 
  [5] evaluate_0.20   rlang_1.0.6     stringi_1.7.12  cli_3.6.0      
- [9] renv_0.16.0-53  vctrs_0.5.1     rmarkdown_2.20  tools_4.2.2    
-[13] stringr_1.5.0   glue_1.6.2      xfun_0.36       yaml_2.3.6     
-[17] fastmap_1.1.0   compiler_4.2.2  htmltools_0.5.4 knitr_1.41     </code></pre>
+ [9] renv_0.16.0-53  rstudioapi_0.14 vctrs_0.5.1     rmarkdown_2.20 
+[13] tools_4.2.2     stringr_1.5.0   glue_1.6.2      xfun_0.36      
+[17] yaml_2.3.6      fastmap_1.1.0   compiler_4.2.2  htmltools_0.5.4
+[21] knitr_1.41     </code></pre>
+</div>
+</div>
+<div class="cell">
+<div class="sourceCode cell-code" id="cb33"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stderr">
+<pre><code>── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
+✔ ggplot2 3.4.0      ✔ purrr   1.0.1 
+✔ tibble  3.1.8      ✔ dplyr   1.0.10
+✔ tidyr   1.2.1      ✔ stringr 1.5.0 
+✔ readr   2.1.3      ✔ forcats 0.5.2 
+── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
+✖ dplyr::filter() masks stats::filter()
+✖ dplyr::lag()    masks stats::lag()</code></pre>
+</div>
+<div class="sourceCode cell-code" id="cb35"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sapply</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">100</span>, <span class="cf">function</span>(i) {</span>
+<span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a>    x <span class="ot">&lt;-</span> <span class="fu">generate_matrix</span>(<span class="dv">100</span>)</span>
+<span id="cb35-3"><a href="#cb35-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">row_wise_scan</span>(x) <span class="sc">==</span> <span class="fu">col_wise_scan</span>(x)</span>
+<span id="cb35-4"><a href="#cb35-4" aria-hidden="true" tabindex="-1"></a>}) <span class="sc">%&gt;%</span> sum <span class="sc">==</span> <span class="dv">100</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
+<div class="cell-output cell-output-stdout">
+<pre><code>[1] TRUE</code></pre>
 </div>
 </div>
 </section>
 
 
-<div id="quarto-appendix" class="default"><section id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes"><h2 class="anchored quarto-appendix-heading">Footnotes</h2>
+<div id="quarto-appendix" class="default"><section class="footnotes footnotes-end-of-document" role="doc-endnotes"><h2 class="anchored quarto-appendix-heading">Footnotes</h2>
 
 <ol>
-<li id="fn1"><p>If your code is right, the following code should evaluate to be <code>TRUE</code></p>
-<div class="sourceCode" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sapply</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">100</span>, <span class="cf">function</span>(i) {</span>
-<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>    x <span class="ot">&lt;-</span> <span class="fu">generate_matrix</span>(<span class="dv">100</span>)</span>
-<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a>    <span class="fu">row_wise_scan</span>(x) <span class="sc">==</span> <span class="fu">col_wise_scan</span>(x)</span>
-<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a>}) <span class="sc">%&gt;%</span> sum <span class="sc">==</span> <span class="dv">100</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
-<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></li>
+<li id="fn1" role="doc-endnote"><p>If your code is right, the following code should evaluate to be <code>TRUE</code><a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
 </ol>
 </section></div></main>
 <!-- /main column -->
@@ -476,24 +595,7 @@ <h1>Appendix</h1>
     button.classList.add('code-copy-button-checked');
     var currentTitle = button.getAttribute("title");
     button.setAttribute("title", "Copied!");
-    let tooltip;
-    if (window.bootstrap) {
-      button.setAttribute("data-bs-toggle", "tooltip");
-      button.setAttribute("data-bs-placement", "left");
-      button.setAttribute("data-bs-title", "Copied!");
-      tooltip = new bootstrap.Tooltip(button, 
-        { trigger: "manual", 
-          customClass: "code-copy-button-tooltip",
-          offset: [0, -8]});
-      tooltip.show();    
-    }
     setTimeout(function() {
-      if (tooltip) {
-        tooltip.hide();
-        button.removeAttribute("data-bs-title");
-        button.removeAttribute("data-bs-toggle");
-        button.removeAttribute("data-bs-placement");
-      }
       button.setAttribute("title", currentTitle);
       button.classList.remove('code-copy-button-checked');
     }, 1000);
@@ -521,50 +623,31 @@ <h1>Appendix</h1>
   for (var i=0; i<noterefs.length; i++) {
     const ref = noterefs[i];
     tippyHover(ref, function() {
-      // use id or data attribute instead here
-      let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
+      let href = ref.getAttribute('href');
       try { href = new URL(href).hash; } catch {}
       const id = href.replace(/^#\/?/, "");
       const note = window.document.getElementById(id);
       return note.innerHTML;
     });
   }
-  const findCites = (el) => {
-    const parentEl = el.parentElement;
-    if (parentEl) {
-      const cites = parentEl.dataset.cites;
-      if (cites) {
-        return {
-          el,
-          cites: cites.split(' ')
-        };
-      } else {
-        return findCites(el.parentElement)
-      }
-    } else {
-      return undefined;
-    }
-  };
   var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]');
   for (var i=0; i<bibliorefs.length; i++) {
     const ref = bibliorefs[i];
-    const citeInfo = findCites(ref);
-    if (citeInfo) {
-      tippyHover(citeInfo.el, function() {
-        var popup = window.document.createElement('div');
-        citeInfo.cites.forEach(function(cite) {
-          var citeDiv = window.document.createElement('div');
-          citeDiv.classList.add('hanging-indent');
-          citeDiv.classList.add('csl-entry');
-          var biblioDiv = window.document.getElementById('ref-' + cite);
-          if (biblioDiv) {
-            citeDiv.innerHTML = biblioDiv.innerHTML;
-          }
-          popup.appendChild(citeDiv);
-        });
-        return popup.innerHTML;
+    const cites = ref.parentNode.getAttribute('data-cites').split(' ');
+    tippyHover(ref, function() {
+      var popup = window.document.createElement('div');
+      cites.forEach(function(cite) {
+        var citeDiv = window.document.createElement('div');
+        citeDiv.classList.add('hanging-indent');
+        citeDiv.classList.add('csl-entry');
+        var biblioDiv = window.document.getElementById('ref-' + cite);
+        if (biblioDiv) {
+          citeDiv.innerHTML = biblioDiv.innerHTML;
+        }
+        popup.appendChild(citeDiv);
       });
-    }
+      return popup.innerHTML;
+    });
   }
 });
 </script>
diff --git a/index.pdf b/index.pdf
index d1729c4..e206d04 100644
Binary files a/index.pdf and b/index.pdf differ
diff --git a/index.qmd b/index.qmd
index 8fd36ac..69dfbab 100644
--- a/index.qmd
+++ b/index.qmd
@@ -1,10 +1,10 @@
 ---
 title: "Homework 1"
-author: "[Insert your name here]{style='background-color: yellow;'}"
+author: "[Avi Singh]{style='background-color: yellow;'}"
 toc: true
 title-block-banner: true
 title-block-style: default
-# format: html
+#format: html
 format: pdf
 ---
 
@@ -64,7 +64,7 @@ In this question, we will walk through the process of _forking_ a `git` reposito
     ```
     Follow the instrutions in order to make sure that `renv` is configured correctly. 
 
-4. Work on the _reminaing part_ of this assignment as a `.qmd` file.
+4. Work on the _remaining part_ of this assignment as a `.qmd` file.
 
     - Create a `PDF` and `HTML` file for your output by modifying the YAML frontmatter for the Quarto `.qmd` document
 
@@ -118,6 +118,20 @@ For the following questions, provide your answers in a code cell.
 
 1. Sort the values of `my_vec_double` in ascending order. 
 
+```{r}
+typeof(my_vec) # 1. The data type is character
+
+my_vec_double <- as.double(my_vec) # 2. Converting data types
+my_vec_int <- as.integer(my_vec)
+
+typeof(my_vec_double) # Verifying the types of vectors are what we want
+typeof(my_vec_int)
+
+my_vec_bool <- ifelse(my_vec_double >= 0, FALSE, TRUE)
+summary(my_vec_bool)["FALSE"] # 3. Count number of False occurrences (4)
+my_vec_double_sorted = sort(my_vec_double) # 4. Sort values of my_vec_double
+```
+
 
 <br><br><br><br>
 <br><br><br><br>
@@ -144,6 +158,13 @@ $$
 \end{bmatrix}
 $$
 
+```{r}
+mat1 <- matrix(c(1,2,3,4,5,6,7,8,9),nrow=3, ncol=3, byrow = TRUE) # Create first matrix
+mat2row1 <- 1:100
+mat2row2 <- (mat2row1)^2
+mat2data <- c(mat2row1, mat2row2)
+mat2 <- matrix(mat2data, nrow = 2, ncol = 100, byrow = TRUE) # Create second matrix
+```
 ::: {.callout-warning}
 ## Tip
 
@@ -180,16 +201,16 @@ mean(M)
 
 2. Write a function `row_wise_scan` which scans the entries of `M` one row after another and outputs the number of elements whose value is $\ge 0$. You can use the following **starter code**
 
-```R
+```{R}
 row_wise_scan <- function(x){
     n <- nrow(x)
     m <- ncol(x)
 
     # Insert your code here
     count <- 0
-    for(...){
-        for(...){
-            if(...){
+    for(i in 1:n){
+        for(value in x[i,]){
+            if(value>=0){
                 count <- count + 1 
             }
         }
@@ -202,11 +223,17 @@ row_wise_scan <- function(x){
 
 3. Similarly, write a function `col_wise_scan` which does exactly the same thing but scans the entries of `M` one column after another
 
-```R
+```{R}
 col_wise_scan <- function(x){
     count <- 0
-    
-    ... # Insert your code here
+    m <- ncol(x)
+    for(i in 1:m){
+      for(value in x[,i]){
+        if(value>=0){
+          count <- count + 1
+        }
+      }
+    }
 
     return(count)
 }
@@ -215,6 +242,8 @@ You can check if your code is doing what it's supposed to using the function her
 
 4. Between `col_wise_scan` and `row_wise_scan`, which function do you expect to take shorter to run? Why?
 
+* I would expect row_wise_scan to take shorter to run since R handles row-wise operations faster than column-wise.
+
 5. Write a function `time_scan` which takes in a method `f` and a matrix `M` and outputs the amount of time taken to run `f(M)`
 
 ```R
@@ -227,24 +256,56 @@ time_scan <- function(f, M){
     return(total_time_taken)
 }
 ```
+```{R}
+time_scan <- function(f, M){
+    initial_time <- Sys.time()
+    f(M)
+    final_time <- Sys.time()
+    
+    total_time_taken <- final_time - initial_time
+    return(total_time_taken)
+}
+```
 
 Provide your output to
 
-```R
+```{R}
 list(
     row_wise_time = time_scan(row_wise_scan, M),
-    col_wise_time = time_scan(row_wise_scan, M)
+    col_wise_time = time_scan(col_wise_scan, M)
 )
 ```
 Which took longer to run? 
 
+row_wise_time took longer to run
+
 6. Repeat this experiment now when:
     * `M` is a $100 \times 100$ matrix
     * `M` is a $1000 \times 1000$ matrix
     * `M` is a $5000 \times 5000$ matrix
+```{r}
+M_100 <- generate_matrix(100)
+M_1000 <- generate_matrix(1000)
+M_5000 <- generate_matrix(5000)
+list(
+    row_wise_time = time_scan(row_wise_scan, M_100),
+    col_wise_time = time_scan(col_wise_scan, M_100)
+    )
+list(
+    row_wise_time = time_scan(row_wise_scan, M_1000),
+    col_wise_time = time_scan(col_wise_scan, M_1000)
+    )
+list(
+    row_wise_time = time_scan(row_wise_scan, M_5000),
+    col_wise_time = time_scan(col_wise_scan, M_5000)
+    )
+
+```
 
 What can you conclude?
 
+* Generally speaking, row_wise_time takes longer to run, and the amount of time it takes to run increases with dimensions of M
+
 <br><br><br><br>
 <br><br><br><br>
 ---
@@ -259,9 +320,10 @@ sessionInfo()
 
 [^footnote]: If your code is right, the following code should evaluate to be `TRUE`
 
-    ```R
-    sapply(1:100, function(i) {
-        x <- generate_matrix(100)
-        row_wise_scan(x) == col_wise_scan(x)
-    }) %>% sum == 100
-    ```
+```{R}
+library(tidyverse)
+sapply(1:100, function(i) {
+    x <- generate_matrix(100)
+    row_wise_scan(x) == col_wise_scan(x)
+}) %>% sum == 100
+```
diff --git a/index_files/libs/bootstrap/bootstrap.min.css b/index_files/libs/bootstrap/bootstrap.min.css
index 62b4dbc..2b6003a 100644
--- a/index_files/libs/bootstrap/bootstrap.min.css
+++ b/index_files/libs/bootstrap/bootstrap.min.css
@@ -3,8 +3,8 @@
  * Copyright 2011-2021 The Bootstrap Authors
  * Copyright 2011-2021 Twitter, Inc.
  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
- */:root{--bs-blue: #0d6efd;--bs-indigo: #6610f2;--bs-purple: #6f42c1;--bs-pink: #d63384;--bs-red: #dc3545;--bs-orange: #fd7e14;--bs-yellow: #ffc107;--bs-green: #198754;--bs-teal: #20c997;--bs-cyan: #0dcaf0;--bs-white: #ffffff;--bs-gray: #6c757d;--bs-gray-dark: #343a40;--bs-gray-100: #f8f9fa;--bs-gray-200: #e9ecef;--bs-gray-300: #dee2e6;--bs-gray-400: #ced4da;--bs-gray-500: #adb5bd;--bs-gray-600: #6c757d;--bs-gray-700: #495057;--bs-gray-800: #343a40;--bs-gray-900: #212529;--bs-default: #dee2e6;--bs-primary: #0d6efd;--bs-secondary: #6c757d;--bs-success: #198754;--bs-info: #0dcaf0;--bs-warning: #ffc107;--bs-danger: #dc3545;--bs-light: #f8f9fa;--bs-dark: #212529;--bs-default-rgb: 222, 226, 230;--bs-primary-rgb: 13, 110, 253;--bs-secondary-rgb: 108, 117, 125;--bs-success-rgb: 25, 135, 84;--bs-info-rgb: 13, 202, 240;--bs-warning-rgb: 255, 193, 7;--bs-danger-rgb: 220, 53, 69;--bs-light-rgb: 248, 249, 250;--bs-dark-rgb: 33, 37, 41;--bs-white-rgb: 255, 255, 255;--bs-black-rgb: 0, 0, 0;--bs-body-color-rgb: 33, 37, 41;--bs-body-bg-rgb: 255, 255, 255;--bs-font-sans-serif: system-ui, -apple-system, "Segoe UI", Roboto, "Helvetica Neue", Arial, "Noto Sans", "Liberation Sans", sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji";--bs-font-monospace: SFMono-Regular, Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace;--bs-gradient: linear-gradient(180deg, rgba(255, 255, 255, 0.15), rgba(255, 255, 255, 0));--bs-root-font-size: 18px;--bs-body-font-family: var(--bs-font-sans-serif);--bs-body-font-size: 1rem;--bs-body-font-weight: 400;--bs-body-line-height: 1.5;--bs-body-color: #212529;--bs-body-bg: #ffffff}*,*::before,*::after{box-sizing:border-box}:root{font-size:var(--bs-root-font-size)}body{margin:0;font-family:var(--bs-body-font-family);font-size:var(--bs-body-font-size);font-weight:var(--bs-body-font-weight);line-height:var(--bs-body-line-height);color:var(--bs-body-color);text-align:var(--bs-body-text-align);background-color:var(--bs-body-bg);-webkit-text-size-adjust:100%;-webkit-tap-highlight-color:rgba(0,0,0,0)}hr{margin:1rem 0;color:inherit;background-color:currentColor;border:0;opacity:.25}hr:not([size]){height:1px}h6,.h6,h5,.h5,h4,.h4,h3,.h3,h2,.h2,h1,.h1{margin-top:0;margin-bottom:.5rem;font-weight:500;line-height:1.2}h1,.h1{font-size:calc(1.345rem + 1.14vw)}@media(min-width: 1200px){h1,.h1{font-size:2.2rem}}h2,.h2{font-size:calc(1.3rem + 0.6vw)}@media(min-width: 1200px){h2,.h2{font-size:1.75rem}}h3,.h3{font-size:calc(1.275rem + 0.3vw)}@media(min-width: 1200px){h3,.h3{font-size:1.5rem}}h4,.h4{font-size:1.25rem}h5,.h5{font-size:1.1rem}h6,.h6{font-size:1rem}p{margin-top:0;margin-bottom:1rem}abbr[title],abbr[data-bs-original-title]{text-decoration:underline dotted;-webkit-text-decoration:underline dotted;-moz-text-decoration:underline dotted;-ms-text-decoration:underline dotted;-o-text-decoration:underline dotted;cursor:help;text-decoration-skip-ink:none}address{margin-bottom:1rem;font-style:normal;line-height:inherit}ol,ul{padding-left:2rem}ol,ul,dl{margin-top:0;margin-bottom:1rem}ol ol,ul ul,ol ul,ul ol{margin-bottom:0}dt{font-weight:700}dd{margin-bottom:.5rem;margin-left:0}blockquote{margin:0 0 1rem;padding:.625rem 1.25rem;border-left:.25rem solid #e9ecef}blockquote p:last-child,blockquote ul:last-child,blockquote ol:last-child{margin-bottom:0}b,strong{font-weight:bolder}small,.small{font-size:0.875em}mark,.mark{padding:.2em;background-color:#fcf8e3}sub,sup{position:relative;font-size:0.75em;line-height:0;vertical-align:baseline}sub{bottom:-0.25em}sup{top:-0.5em}a{color:#0d6efd;text-decoration:underline;-webkit-text-decoration:underline;-moz-text-decoration:underline;-ms-text-decoration:underline;-o-text-decoration:underline}a:hover{color:#0a58ca}a:not([href]):not([class]),a:not([href]):not([class]):hover{color:inherit;text-decoration:none}pre,code,kbd,samp{font-family:var(--bs-font-monospace);font-size:1em;direction:ltr /* rtl:ignore */;unicode-bidi:bidi-override}pre{display:block;margin-top:0;margin-bottom:1rem;overflow:auto;font-size:0.875em;color:#000;background-color:#f6f6f6;padding:.5rem;border:1px solid #dee2e6;border-radius:.25rem}pre code{background-color:transparent;font-size:inherit;color:inherit;word-break:normal}code{font-size:0.875em;color:#9753b8;background-color:#f6f6f6;border-radius:.25rem;padding:.125rem .25rem;word-wrap:break-word}a>code{color:inherit}kbd{padding:.4rem .4rem;font-size:0.875em;color:#fff;background-color:#212529;border-radius:.2rem}kbd kbd{padding:0;font-size:1em;font-weight:700}figure{margin:0 0 1rem}img,svg{vertical-align:middle}table{caption-side:bottom;border-collapse:collapse}caption{padding-top:.5rem;padding-bottom:.5rem;color:#6c757d;text-align:left}th{text-align:inherit;text-align:-webkit-match-parent}thead,tbody,tfoot,tr,td,th{border-color:inherit;border-style:solid;border-width:0}label{display:inline-block}button{border-radius:0}button:focus:not(:focus-visible){outline:0}input,button,select,optgroup,textarea{margin:0;font-family:inherit;font-size:inherit;line-height:inherit}button,select{text-transform:none}[role=button]{cursor:pointer}select{word-wrap:normal}select:disabled{opacity:1}[list]::-webkit-calendar-picker-indicator{display:none}button,[type=button],[type=reset],[type=submit]{-webkit-appearance:button}button:not(:disabled),[type=button]:not(:disabled),[type=reset]:not(:disabled),[type=submit]:not(:disabled){cursor:pointer}::-moz-focus-inner{padding:0;border-style:none}textarea{resize:vertical}fieldset{min-width:0;padding:0;margin:0;border:0}legend{float:left;width:100%;padding:0;margin-bottom:.5rem;font-size:calc(1.275rem + 0.3vw);line-height:inherit}@media(min-width: 1200px){legend{font-size:1.5rem}}legend+*{clear:left}::-webkit-datetime-edit-fields-wrapper,::-webkit-datetime-edit-text,::-webkit-datetime-edit-minute,::-webkit-datetime-edit-hour-field,::-webkit-datetime-edit-day-field,::-webkit-datetime-edit-month-field,::-webkit-datetime-edit-year-field{padding:0}::-webkit-inner-spin-button{height:auto}[type=search]{outline-offset:-2px;-webkit-appearance:textfield}::-webkit-search-decoration{-webkit-appearance:none}::-webkit-color-swatch-wrapper{padding:0}::file-selector-button{font:inherit}::-webkit-file-upload-button{font:inherit;-webkit-appearance:button}output{display:inline-block}iframe{border:0}summary{display:list-item;cursor:pointer}progress{vertical-align:baseline}[hidden]{display:none !important}.lead{font-size:1.25rem;font-weight:300}.display-1{font-size:calc(1.625rem + 4.5vw);font-weight:300;line-height:1.2}@media(min-width: 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diff --git a/index_files/libs/quarto-html/quarto.js b/index_files/libs/quarto-html/quarto.js
index e62a8cb..c7e1c84 100644
--- a/index_files/libs/quarto-html/quarto.js
+++ b/index_files/libs/quarto-html/quarto.js
@@ -6,7 +6,7 @@ const sectionChanged = new CustomEvent("quarto-sectionChanged", {
 });
 
 window.document.addEventListener("DOMContentLoaded", function (_event) {
-  const tocEl = window.document.querySelector('nav.toc-active[role="doc-toc"]');
+  const tocEl = window.document.querySelector('nav[role="doc-toc"]');
   const sidebarEl = window.document.getElementById("quarto-sidebar");
   const leftTocEl = window.document.getElementById("quarto-sidebar-toc-left");
   const marginSidebarEl = window.document.getElementById(
@@ -33,9 +33,6 @@ window.document.addEventListener("DOMContentLoaded", function (_event) {
     tab.addEventListener("shown.bs.tab", fireSlideEnter);
   });
 
-  // fire slideEnter for tabby tab activations (for htmlwidget resize behavior)
-  document.addEventListener("tabby", fireSlideEnter, false);
-
   // Track scrolling and mark TOC links as active
   // get table of contents and sidebar (bail if we don't have at least one)
   const tocLinks = tocEl
@@ -278,7 +275,7 @@ window.document.addEventListener("DOMContentLoaded", function (_event) {
         const convertToMenu = () => {
           for (const child of el.children) {
             child.style.opacity = 0;
-            child.style.overflow = "hidden";
+            child.style.display = "none";
           }
 
           const toggleContainer = window.document.createElement("div");
@@ -381,7 +378,7 @@ window.document.addEventListener("DOMContentLoaded", function (_event) {
         const convertToSidebar = () => {
           for (const child of el.children) {
             child.style.opacity = 1;
-            child.style.overflow = null;
+            clone.style.display = null;
           }
 
           const placeholderEl = window.document.getElementById(
@@ -423,21 +420,18 @@ window.document.addEventListener("DOMContentLoaded", function (_event) {
   const marginChildren = window.document.querySelectorAll(
     ".column-margin.column-container > * "
   );
-
-  nexttick(() => {
-    let lastBottom = 0;
-    for (const marginChild of marginChildren) {
-      const top = marginChild.getBoundingClientRect().top + window.scrollY;
-      if (top < lastBottom) {
-        const margin = lastBottom - top;
-        marginChild.style.marginTop = `${margin}px`;
-      }
-      const styles = window.getComputedStyle(marginChild);
-      const marginTop = parseFloat(styles["marginTop"]);
-
-      lastBottom = top + marginChild.getBoundingClientRect().height + marginTop;
+  let lastBottom = 0;
+  for (const marginChild of marginChildren) {
+    const top = marginChild.getBoundingClientRect().top;
+    if (top < lastBottom) {
+      const margin = lastBottom - top;
+      marginChild.style.marginTop = `${margin}px`;
     }
-  });
+    const styles = window.getComputedStyle(marginChild);
+    const marginTop = parseFloat(styles["marginTop"]);
+
+    lastBottom = top + marginChild.getBoundingClientRect().height + marginTop;
+  }
 
   // Manage the visibility of the toc and the sidebar
   const marginScrollVisibility = manageSidebarVisiblity(marginSidebarEl, {
@@ -659,99 +653,6 @@ window.document.addEventListener("DOMContentLoaded", function (_event) {
   highlightReaderToggle(isReaderMode());
 });
 
-// grouped tabsets
-window.addEventListener("pageshow", (_event) => {
-  function getTabSettings() {
-    const data = localStorage.getItem("quarto-persistent-tabsets-data");
-    if (!data) {
-      localStorage.setItem("quarto-persistent-tabsets-data", "{}");
-      return {};
-    }
-    if (data) {
-      return JSON.parse(data);
-    }
-  }
-
-  function setTabSettings(data) {
-    localStorage.setItem(
-      "quarto-persistent-tabsets-data",
-      JSON.stringify(data)
-    );
-  }
-
-  function setTabState(groupName, groupValue) {
-    const data = getTabSettings();
-    data[groupName] = groupValue;
-    setTabSettings(data);
-  }
-
-  function toggleTab(tab, active) {
-    const tabPanelId = tab.getAttribute("aria-controls");
-    const tabPanel = document.getElementById(tabPanelId);
-    if (active) {
-      tab.classList.add("active");
-      tabPanel.classList.add("active");
-    } else {
-      tab.classList.remove("active");
-      tabPanel.classList.remove("active");
-    }
-  }
-
-  function toggleAll(selectedGroup, selectorsToSync) {
-    for (const [thisGroup, tabs] of Object.entries(selectorsToSync)) {
-      const active = selectedGroup === thisGroup;
-      for (const tab of tabs) {
-        toggleTab(tab, active);
-      }
-    }
-  }
-
-  function findSelectorsToSyncByLanguage() {
-    const result = {};
-    const tabs = Array.from(
-      document.querySelectorAll(`div[data-group] a[id^='tabset-']`)
-    );
-    for (const item of tabs) {
-      const div = item.parentElement.parentElement.parentElement;
-      const group = div.getAttribute("data-group");
-      if (!result[group]) {
-        result[group] = {};
-      }
-      const selectorsToSync = result[group];
-      const value = item.innerHTML;
-      if (!selectorsToSync[value]) {
-        selectorsToSync[value] = [];
-      }
-      selectorsToSync[value].push(item);
-    }
-    return result;
-  }
-
-  function setupSelectorSync() {
-    const selectorsToSync = findSelectorsToSyncByLanguage();
-    Object.entries(selectorsToSync).forEach(([group, tabSetsByValue]) => {
-      Object.entries(tabSetsByValue).forEach(([value, items]) => {
-        items.forEach((item) => {
-          item.addEventListener("click", (_event) => {
-            setTabState(group, value);
-            toggleAll(value, selectorsToSync[group]);
-          });
-        });
-      });
-    });
-    return selectorsToSync;
-  }
-
-  const selectorsToSync = setupSelectorSync();
-  for (const [group, selectedName] of Object.entries(getTabSettings())) {
-    const selectors = selectorsToSync[group];
-    // it's possible that stale state gives us empty selections, so we explicitly check here.
-    if (selectors) {
-      toggleAll(selectedName, selectors);
-    }
-  }
-});
-
 function throttle(func, wait) {
   let waiting = false;
   return function () {
@@ -764,7 +665,3 @@ function throttle(func, wait) {
     }
   };
 }
-
-function nexttick(func) {
-  return setTimeout(func, 0);
-}
