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LE1 move matrix definition to right before where it's needed #559

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136 changes: 69 additions & 67 deletions source/linear-algebra/source/01-LE/01.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -58,70 +58,6 @@
</activity>
</subsection>
<subsection><title>Class Activities</title>

<definition>
<statement>
<p>A <term>matrix</term><idx>matrix</idx> is an <m>m\times n</m> array of real numbers
with <m>m</m> rows and <m>n</m> columns:
<me>
\left[\begin{array}{cccc}
a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\
a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\
\vdots &amp; \vdots &amp; \ddots &amp; \vdots \\
a_{m1} &amp; a_{m2} &amp; \cdots &amp; a_{mn} \\
\end{array}\right]
=
\left[\begin{array}{cccc} \vec v_1 &amp; \vec v_2 &amp; \cdots &amp; \vec v_n\end{array}\right]
</me>.
Frequently we will use matrices to describe an ordered list of
its <term>column vectors</term>:
<me>
\left[\begin{array}{c}
a_{11} \\
a_{21} \\
\vdots \\
a_{m1} \\
\end{array}\right],
\left[\begin{array}{c}
a_{12} \\
a_{22} \\
\vdots \\
a_{m2} \\
\end{array}\right],\cdots,
\left[\begin{array}{c}
a_{1n} \\
a_{2n} \\
\vdots \\
a_{mn} \\
\end{array}\right] =
\vec v_1, \vec v_2, \cdots, \vec v_n
</me>.
When order is irrelevant, we will use set notation:
<me>
\left\{
\left[\begin{array}{c}
a_{11} \\
a_{21} \\
\vdots \\
a_{m1} \\
\end{array}\right],
\left[\begin{array}{c}
a_{12} \\
a_{22} \\
\vdots \\
a_{m2} \\
\end{array}\right],\cdots,
\left[\begin{array}{c}
a_{1n} \\
a_{2n} \\
\vdots \\
a_{mn} \\
\end{array}\right]\right\} =
\{\vec v_1, \vec v_2, \cdots, \vec v_n\}
</me>.
</p>
</statement>
</definition>
<definition>
<statement>
<p>
Expand All @@ -135,9 +71,12 @@ list of real numbers
a_n
\end{array}\right]
</me>.
We will find it useful to almost always typeset Euclidean vectors vertically, but the notation
<m>\left[\begin{array}{cccc}a_1 &amp; a_2 &amp; \cdots &amp; a_n\end{array}\right]^T</m> is also
valid when vertical typesetting is inconvenient. The set of all Euclidean vectors with
The notations <m>(a_1, a_2, \cdots, a_n)</m> and <m>\langle a_1, a_2, \cdots, a_n)</m>
are used in other mathematical contexts, but
we will find it useful to almost always typeset Euclidean vectors vertically.
However, the notation
<me>\left[\begin{array}{cccc}a_1 &amp; a_2 &amp; \cdots &amp; a_n\end{array}\right]^T</me>
is also valid when vertical typesetting is inconvenient. The set of all Euclidean vectors with
<m>n</m> components is denoted as <m>\mathbb R^n</m>, and vectors are often described using
the notation <m>\vec v</m>.
</p>
Expand Down Expand Up @@ -687,6 +626,69 @@ Then use these to describe the solution set
</sidebyside>
</remark>

<definition>
<statement>
<p>A <term>matrix</term><idx>matrix</idx> is an <m>m\times n</m> array of real numbers
with <m>m</m> rows and <m>n</m> columns:
<me>
\left[\begin{array}{cccc}
a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\
a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\
\vdots &amp; \vdots &amp; \ddots &amp; \vdots \\
a_{m1} &amp; a_{m2} &amp; \cdots &amp; a_{mn} \\
\end{array}\right]
=
\left[\begin{array}{cccc} \vec v_1 &amp; \vec v_2 &amp; \cdots &amp; \vec v_n\end{array}\right]
</me>.
Frequently we will use matrices to describe an ordered list of
<term>column vectors</term>:
<me>
\left[\begin{array}{c}
a_{11} \\
a_{21} \\
\vdots \\
a_{m1} \\
\end{array}\right],
\left[\begin{array}{c}
a_{12} \\
a_{22} \\
\vdots \\
a_{m2} \\
\end{array}\right],\cdots,
\left[\begin{array}{c}
a_{1n} \\
a_{2n} \\
\vdots \\
a_{mn} \\
\end{array}\right] =
\vec v_1, \vec v_2, \cdots, \vec v_n
</me>.
When order is irrelevant, we will use set notation:
<me>
\left\{
\left[\begin{array}{c}
a_{11} \\
a_{21} \\
\vdots \\
a_{m1} \\
\end{array}\right],
\left[\begin{array}{c}
a_{12} \\
a_{22} \\
\vdots \\
a_{m2} \\
\end{array}\right],\cdots,
\left[\begin{array}{c}
a_{1n} \\
a_{2n} \\
\vdots \\
a_{mn} \\
\end{array}\right]\right\} =
\{\vec v_1, \vec v_2, \cdots, \vec v_n\}
</me>.
</p>
</statement>
</definition>
<definition>
<statement>
<p>
Expand Down
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