Skip to content

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

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Draft
StevenClontz wants to merge 1 commit into
base: main
Choose a base branch
from
Draft

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

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
136 changes: 69 additions & 67 deletions source/linear-algebra/source/01-LE/01.ptx
View file
Open in desktop
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
Loading