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Rearranged handout

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Mark
2023-04-03 11:13:33 -07:00
parent ef6fa1da2b
commit ef0abb5f17
5 changed files with 24 additions and 27 deletions
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35
Advanced/Linear Maps/parts/3 matrices.tex
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@ -17,8 +17,8 @@ Draw a $3 \times 2$ matrix.
\vfill
\definition{}
We can define the \say{product\footnotemark{}} of a matrix $A$ and a vector $v$:
\footnotetext{This is an uncommon word to use in this context. You will soon see why.}
We can define the product of a matrix $A$ and a vector $v$:
$$
Av =
\begin{bmatrix}
@ -34,7 +34,7 @@ Av =
4a + 5b + 6c
\end{bmatrix}
$$
Look closely. Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
$$
Av =
@ -61,9 +61,9 @@ Compute the following:
$$
\begin{bmatrix}
2 & 9 \\
7 & 5 \\
3 & 4
1 & 2 \\
3 & 4 \\
5 & 6
\end{bmatrix}
\begin{bmatrix}
5 \\ 3
@ -84,16 +84,16 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol
$$
\begin{bmatrix}
2 & 9 \\
7 & 5 \\
3 & 4
1 & 2 \\
3 & 4 \\
5 & 6
\end{bmatrix}
\begin{bmatrix}
5 \\ 3
\end{bmatrix}
=
\begin{bmatrix}
37 \\ 50 \\ 27
11 \\ 27 \\ 43
\end{bmatrix}
$$
\end{center}
@ -110,9 +110,9 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol
left delimiter={[},
right delimiter={]}
] (A) {
2 & 9 \\
7 & 5 \\
1 & 2 \\
3 & 4 \\
5 & 6 \\
};
\node[
@ -133,21 +133,21 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol
\node[
fit=(A-1-2)(A-1-2),
inner xsep=8mm,inner ysep=0mm,
label=right:{$10 + 27 = 37$}
label=right:{$5 + 6 = 11$}
](Y) {};
\draw[->, gray] ([xshift=3mm]A-1-2.east) -- (Y);
\node[
fit=(A-2-2)(A-2-2),
inner xsep=8mm,inner ysep=0mm,
label=right:{$35 + 15 = 50$}
label=right:{$15 + 12 = 27$}
](H) {};
\draw[->, gray] ([xshift=3mm]A-2-2.east) -- (H);
\node[
fit=(A-3-2)(A-3-2),
inner xsep=8mm,inner ysep=0mm,
label=right:{$15 + 12 = 27$}
label=right:{$25 + 18 = 43$}
](N) {};
\draw[->, gray] ([xshift=3mm]A-3-2.east) -- (N);
\end{tikzpicture}
@ -187,11 +187,6 @@ Before you start, answer the following questions:
Show that any linear transformation can be written as a matrix.
\vfill
\pagebreak
\problem{}
Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space.
\vfill
\problem{}