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Mark 2023-03-26 22:09:51 -07:00
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8
Advanced/Linear Maps/main.tex
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@ -39,12 +39,12 @@
\section{Bonus} \section{Bonus}
\problem{}
Is the set of all linear maps a vector space?
\vfill
\definition{} \definition{}
Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space. Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space.
\vfill \vfill
\problem{}
Is the set of all linear maps a vector space?
\vfill
\end{document} \end{document}

2
Advanced/Linear Maps/parts/2 linearity.tex
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@ -4,7 +4,7 @@
A \textit{function} or \textit{map} $f$ from a set $A$ to a set $B$ is a rule that assigns an element of $B$ to each element of $A$. We write this as $f: A \to B$. A \textit{function} or \textit{map} $f$ from a set $A$ to a set $B$ is a rule that assigns an element of $B$ to each element of $A$. We write this as $f: A \to B$.
\definition{This one is worth remembering} \definition{}
Let $V$ and $U$ be vector spaces, and let $f: V \to U$ be a map from $V$ to $U$. \\ Let $V$ and $U$ be vector spaces, and let $f: V \to U$ be a map from $V$ to $U$. \\
We say $f$ is \textit{linear} if it satisfies the following for any $v \in V$, $u \in U$, $a \in \mathbb{R}$: We say $f$ is \textit{linear} if it satisfies the following for any $v \in V$, $u \in U$, $a \in \mathbb{R}$:
\begin{itemize} \begin{itemize}

20
Advanced/Linear Maps/parts/3 matrices.tex
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@ -16,7 +16,7 @@ Draw a $3 \times 2$ matrix.
\vfill \vfill
\definition{Matrices as Transformations} \definition{}
We can define the \say{product\footnotemark{}} of a matrix $A$ and a vector $v$: 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.} \footnotetext{This is an uncommon word to use in this context. You will soon see why.}
$$ $$
@ -57,7 +57,7 @@ $$
Naturally, a vector can only be multiplied by a matrix if the number of rows in the vector equals the number of columns in the matrix. Naturally, a vector can only be multiplied by a matrix if the number of rows in the vector equals the number of columns in the matrix.
\problem{} \problem{}
Compute the following \say{product}: Compute the following:
$$ $$
\begin{bmatrix} \begin{bmatrix}
@ -186,13 +186,27 @@ Before you start, answer the following questions:
\problem{}<proofback> \problem{}<proofback>
Show that any linear transformation can be written as a matrix. 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{}
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 $D$. \\
\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How solutions?}
\vfill \vfill
\pagebreak \pagebreak
\problem{} \problem{}
Does \ref{thebigtheorem} hold in arbitrary vector spaces? \\ Does \ref{thebigtheorem} hold in arbitrary vector spaces? \\
Repeat \ref{prooffwd} and \ref{proofback} using only axioms. Repeat \ref{prooffwd} and \ref{proofback} using only axioms, without assuming that we're working in $\mathbb{R}^n$.
\vfill \vfill
\pagebreak \pagebreak