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EV3 activities revision #317

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39 changes: 33 additions & 6 deletions source/linear-algebra/source/02-EV/03.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -46,8 +46,11 @@
<activity>
<statement>
<p>
Let <m>S</m> denote a set of vectors in <m>\IR^n</m> and suppose that <m>\vec{u},\vec{v}\in\vspan(S)</m>,
<m>c\in\IR</m> and that <m>\vec{w}\in\IR^n</m>. Which of the following vectors might
Let <m>S=\{\vec v_1,\dots,\vec v_n\}</m> denote a set of vectors in <m>\IR^n</m>.
</p>
<p>Suppose that
<m>\vec{u},\vec{v}\in\vspan(S)</m>,
<m>c\in\IR</m>, and <m>\vec{w}\in\IR^n</m>. Which of the following vectors might
<em>not</em> belong to <m>\vspan(S)</m>?
<ol marker="A.">
<li><m>\vec{0}</m></li>
Expand All @@ -59,6 +62,30 @@
</statement>
</activity>

<remark>
<p>
If <m>S</m> is any set of vectors in <m>\IR^n</m>, then the set <m>\vspan(S)</m> has the following properties:
<ul>
<li>
<p>
the set <m>\vspan(S)</m> is non-empty, specifically, it at least contains <m>\vec 0</m>.
</p>
</li>
<li>
<p>
the set <m>\vspan(S)</m> is closed under addition: for any <m>\vec{u},\vec{v}\in \vspan(S)</m>, the sum <m>\vec{u}+\vec{v}</m> is also in <m>\vspan(S)</m>.
</p>
</li>
<li>
<p>
the set <m>\vspan(S)</m> is closed under scalar multiplication: for any <m>\vec{u}\in\vspan(S)</m> and scalar <m>c\in\IR</m>, the product <m>c\vec{u}</m> is also in <m>\vspan(S)</m>.
</p>
</li>
</ul>
It will be interesting to see if these kinds of properties might hold in other scenarios.
</p>
</remark>

<definition>
<statement>
<p>
Expand Down Expand Up @@ -193,7 +220,7 @@
<ul>
<li>
<p>
the set <m>\vspan(S)</m> is non-empty.
the set <m>\vspan(S)</m> is non-empty, specifically, it at least contains <m>\vec 0</m>.
</p>
</li>
<li>
Expand All @@ -211,17 +238,17 @@
<ul>
<li>
<p>
the set <m>W</m> is non-empty.
the solution set <m>W</m> is non-empty, specifically, it at least contains <m>\vec 0</m>.
</p>
</li>
<li>
<p>
the set <m>W</m> is closed under addition: for any <m>\vec{u},\vec{v}\in W</m>, the sum <m>\vec{u}+\vec{v}</m> is also in <m>W</m>.
the solution set <m>W</m> is closed under addition: for any <m>\vec{u},\vec{v}\in W</m>, the sum <m>\vec{u}+\vec{v}</m> is also in <m>W</m>.
</p>
</li>
<li>
<p>
the set <m>\vspan(S)</m> is closed under scalar multiplication: for any <m>\vec{u}\in W</m> and scalar <m>c\in\IR</m>, the product <m>c\vec{u}</m> is also in <m>W</m>.
the solution set <m>\W</m> is closed under scalar multiplication: for any <m>\vec{u}\in W</m> and scalar <m>c\in\IR</m>, the product <m>c\vec{u}</m> is also in <m>W</m>.
</p>
</li>
</ul>
Expand Down
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