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Mark 2023-04-03 13:19:23 -07:00
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2
Advanced/Linear Maps/main.tex
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@ -44,7 +44,7 @@
\vfill \vfill
\problem{} \problem{}
Is the set of all linear maps a vector space? Show that the set of all linear maps is a vector space.
\vfill \vfill
\end{document} \end{document}

36
Advanced/Linear Maps/parts/3 matrices.tex
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@ -11,14 +11,8 @@ A =
$$ $$
The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix. The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix.
\problem{} We can define the product of a matrix $A$ and a vector $v$ as follows:
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.}
$$ $$
Av = Av =
\begin{bmatrix} \begin{bmatrix}
@ -34,7 +28,7 @@ Av =
4a + 5b + 6c 4a + 5b + 6c
\end{bmatrix} \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 = Av =
@ -56,6 +50,13 @@ $$
Naturally, a vector can only be multiplied by a matrix if the number of rows in the vector equals the number of columns in the matrix. Naturally, a vector can only be multiplied by a matrix if the number of rows in the vector equals the number of columns in the matrix.
\problem{}
Say you multiply a size-$m$ vector by an $m \times n$ matrix. What is the size of your result?
\vfill
\problem{} \problem{}
Compute the following: Compute the following:
@ -156,7 +157,7 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol
\vspace{2mm} \vspace{2mm}
Be aware that this is only a model for intuition. \\ This is only a model for intuition, though. \\
Make sure you understand the dot product definition on the previous page. Make sure you understand the dot product definition on the previous page.
\vspace{5mm} \vspace{5mm}
@ -167,24 +168,15 @@ Conversely, every $n \times m$ matrix represents a liner map $T: \mathbb{R}^n \t
\vspace{2mm} \vspace{2mm}
In other words, \textbf{matrices are linear transformations}. \\ In other words, \textbf{matrices are linear transformations}.
If you only learn only one thing today, this should be it.
\vfill
\problem{}<prooffwd> \problem{}<prooffwd>
Show that the transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\ Show that the transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\
Before you start, answer the following questions: \hint{What is $A$? What is $v$? What are their sizes?}
\begin{itemize}
\item What is $A$?
\item What is $v$?
\item What are their sizes?
\end{itemize}
\vfill \vfill
\problem{}<proofback> \problem{}<proofback>
Show that any linear transformation can be written as a matrix. Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as $T(v) = Av$.
\vfill \vfill
\pagebreak \pagebreak
@ -197,7 +189,7 @@ Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector spac
\problem{} \problem{}
Consider the transformation $D: \mathbb{P}^3 \to \mathbb{P}^2$ defined by $D(p) = \frac{d}{dx}p$. \\ Consider the transformation $D: \mathbb{P}^3 \to \mathbb{P}^2$ defined by $D(p) = \frac{d}{dx}p$. \\
Find a matrix that corresponds to $D$. \\ Find a matrix that corresponds to $D$. \\
\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How solutions?} \hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How so?}
\vfill \vfill