Removed linear map handout
This commit is contained in:
parent
f357725f5b
commit
2e3e1b3c56
@ -1,49 +0,0 @@
|
||||
% use [nosolutions] flag to hide solutions.
|
||||
% use [solutions] flag to show solutions.
|
||||
\documentclass[
|
||||
solutions,
|
||||
nowarning,
|
||||
singlenumbering
|
||||
]{../../resources/ormc_handout}
|
||||
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{
|
||||
matrix,
|
||||
decorations.pathreplacing,
|
||||
calc,
|
||||
positioning,
|
||||
fit
|
||||
}
|
||||
|
||||
|
||||
% Let's give clarifications about the meaning of Z and R when we use them in the first problems.
|
||||
|
||||
% Definitely define $R^n$ before using. Optionally you may add a problem "convince yourself that $R^2$ is a plane and $R^3$ is a 3-d space".
|
||||
|
||||
% Maybe we can add an example of a linear transformation from R^2 to R^2? Rotation? Scaling of y-axis?
|
||||
|
||||
% Slow down, understand linear transformations fully.
|
||||
|
||||
|
||||
|
||||
|
||||
%\usepackage{lua-visual-debug}
|
||||
\renewcommand{\arraystretch}{1.2}
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
<Advanced 2>
|
||||
<Spring 2023>
|
||||
{Linear Maps}
|
||||
{
|
||||
Prepared by Mark on \today \\
|
||||
}
|
||||
|
||||
%\input{parts/? fields}
|
||||
%\input{parts/? spaces}
|
||||
|
||||
\input{parts/0 intro}
|
||||
\input{parts/1 linear}
|
||||
\input{parts/2 matrices}
|
||||
|
||||
\end{document}
|
||||
@ -1,6 +0,0 @@
|
||||
\section{Intro}
|
||||
|
||||
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
@ -1,58 +0,0 @@
|
||||
\section{Linear Maps}
|
||||
|
||||
\definition{}
|
||||
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{}<lineardef>
|
||||
Let $f: U \to V$ be a map. \\
|
||||
We say $f$ is \textit{linear} if it satisfies the following for any $u \in U$, $v \in V$, $a \in \mathbb{R}$:
|
||||
\begin{itemize}
|
||||
\item $f(u + v) = f(u) + f(v)$
|
||||
\item $f(au) = af(u)$
|
||||
\end{itemize}
|
||||
In other words, $f$ is linear if it is \say{closed} under addition and scalar multiplication.
|
||||
|
||||
|
||||
\problem{}
|
||||
It is often convenient to combine the two conditions above into one. \\
|
||||
Show that $f(au + v) = af(u) + f(v)$ iff $f$ is linear. Use \ref{lineardef}.
|
||||
|
||||
\vfill
|
||||
|
||||
\problem{}
|
||||
Is $f(x) = mx + b$ a linear map?
|
||||
|
||||
\vfill
|
||||
|
||||
\problem{}
|
||||
In general, what does a linear map in $\mathbb{R} \to \mathbb{R}$ look like?
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
|
||||
\problem{}
|
||||
Is the map ${median}(v): \mathbb{R}^3 \to \mathbb{R}$ linear? \\
|
||||
\hint{$median([3, 5, 4]) = 4$, but you already knew that.}
|
||||
|
||||
\vfill
|
||||
|
||||
\problem{}
|
||||
Is the map $f(v): \mathbb{R}^3 \to \mathbb{R}$ defined by $f(v) = v_0 + 2v_1 + v_2$ linear? \\
|
||||
\hint{$v_n$ is the $n^\text{th}$ element of $v$. $v$ is a 3-element vector.}
|
||||
|
||||
|
||||
\vfill
|
||||
|
||||
\problem{}
|
||||
Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? \\
|
||||
\vspace{1mm}
|
||||
\hint{$\mathbb{P}^n$ is the set of polynomials with degree at most $n$.}
|
||||
|
||||
\vfill
|
||||
|
||||
\problem{}
|
||||
In general, what does a linear map from $\mathbb{R}^m \to \mathbb{R}^n$ look like?
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
@ -1,34 +0,0 @@
|
||||
\section{Matrices}
|
||||
|
||||
\theorem{}<thebigtheorem>
|
||||
Any linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as an $n \times m$ matrix. \\
|
||||
Conversely, every $n \times m$ matrix represents a linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ \\
|
||||
|
||||
\vspace{2mm}
|
||||
|
||||
In other words, \textbf{matrices are linear transformations}. \\
|
||||
The next two problems provide a proof.
|
||||
|
||||
|
||||
\problem{}<prooffwd>
|
||||
Let $A$ be an $m \times n$ matrix, and $v$ an $m \times 1$ vector. \\
|
||||
Show that the map $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\
|
||||
|
||||
|
||||
|
||||
\vfill
|
||||
|
||||
\problem{}<proofback>
|
||||
Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as $T(v) = Av$.
|
||||
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
|
||||
\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 so?}
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
@ -1,49 +0,0 @@
|
||||
\section{Fields}
|
||||
|
||||
\definition{Fields and Field Axioms}
|
||||
A \textit{field} $\mathbb{F}$ consists of a set $A$ and two operations $+$ and $\times$. \\
|
||||
As usual, we may abbreviate $a \times b$ as $ab$. \\
|
||||
The following axioms must be satisfied for any $a, b, c \in \mathbb{F}$:
|
||||
|
||||
\vspace{1mm}
|
||||
\begin{center}
|
||||
% @{} supresses the space between columns.
|
||||
% @{=} makes = a column seperator.
|
||||
\begin{tabular}{l | r@{=}l | r@{=}l}
|
||||
\hline
|
||||
\multicolumn{1}{|c|}{Name} &
|
||||
\multicolumn{2}{c}{$+$} &
|
||||
\multicolumn{2}{|c|}{$\times$} \\
|
||||
\hline
|
||||
Closure & \multicolumn{2}{c|}{$a+b \in \mathbb{F}$} & \multicolumn{2}{c}{$ab \in \mathbb{F}$} \\
|
||||
Associativity & $(a+b)+c~$&$~a+b+c$ & $(ab)c~$&$~a(bc)$ \\
|
||||
Commutativity & $a+b~$&$~b+a$ & $ab~$&$~ba$ \\
|
||||
Distributivity & $a(b+c)~$&$~ab + ac$ & \multicolumn{2}{}{} \\
|
||||
Identity & $a+0~$&$~a$ & $1 \times a~$&$~a$ \\
|
||||
Inverses & $a + (-a)~$&$~0$ & $a \times a^{-1}~$&$~1$
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
||||
|
||||
\problem{}
|
||||
Show that all fields are groups. \\
|
||||
Convince yourself that not all groups are fields.
|
||||
|
||||
\vfill
|
||||
|
||||
|
||||
\problem{}
|
||||
Is $\mathbb{Z}$ a field under our usual definitions of $+$ and $\times$? \\
|
||||
Which axioms does it satisfy, and which does it violate?
|
||||
|
||||
\vfill
|
||||
|
||||
\problem{}
|
||||
Verify that $\mathbb{R}$ is a field.
|
||||
\vfill
|
||||
|
||||
\generic{Remark:}
|
||||
We won't worry too much about fields this week. They simply provide a foundation for \textit{spaces}. \\
|
||||
As such, you may assume that we are working in $\mathbb{R}$ for the rest of this handout.
|
||||
|
||||
\pagebreak
|
||||
@ -1,113 +0,0 @@
|
||||
\section{Spaces}
|
||||
|
||||
\definition{}
|
||||
A \textit{space} over a field $\mathbb{F}$ consists of the following elements:
|
||||
\begin{itemize}[itemsep = 2mm]
|
||||
\item A set $V$, the elements of which are called \textit{vectors}
|
||||
\item An operation called \textit{vector addition}, denoted $+$ \\
|
||||
Vector addition operates on two elements of $V$. \\
|
||||
|
||||
\item An operation called \textit{scalar multiplication}, denoted $\times$ \\
|
||||
Scalar multiplication multiplies an element of $V$ by an element of $\mathbb{F}$. \\
|
||||
Any element of $\mathbb{F}$ is called a \textit{scalar}.
|
||||
\end{itemize}
|
||||
|
||||
\vspace{2mm}
|
||||
|
||||
\textbf{Note:}
|
||||
The same symbols are used for additions and multiplications in both $\mathbb{F}$ and $V$. \\
|
||||
\textit{These are different operations}, so be aware of the context of each $+$ and $\times$.
|
||||
|
||||
\vspace{5mm}
|
||||
|
||||
Vector addition and multiplication must have the following properties. \\
|
||||
In both tables, $x, y, z \in V$ and $a, b\in \mathbb{F}$.
|
||||
|
||||
\vspace{2mm}
|
||||
|
||||
% [t] and \vspace{0pt} ensure alignment at top
|
||||
\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
|
||||
\begin{center}
|
||||
\begin{tabular}{l | r@{=}l }
|
||||
\hline
|
||||
\multicolumn{3}{|c|}{Properties of vector addition} \\
|
||||
\hline
|
||||
Closure & \multicolumn{2}{c}{$x+y \in V$} \\
|
||||
Associativity & $(x+y)+z~$&$~x+y+z$ \\
|
||||
Commutativity & $x+y~$&$~y+x$ \\
|
||||
Distributivity & $x(y+z)~$&$~xy + xz$ \\
|
||||
Identity & $x+0~$&$~x$ \\
|
||||
Inverse & $x + (-x)~$&$~0$
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\end{minipage}%
|
||||
\hfill%
|
||||
\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
|
||||
\begin{center}
|
||||
\begin{tabular}{l | r@{=}l }
|
||||
\hline
|
||||
\multicolumn{3}{|c|}{Properties of scalar multiplication} \\
|
||||
\hline
|
||||
Closure & \multicolumn{2}{c}{$ax \in V$} \\
|
||||
Distributivity & $a(x+y)~$&$~ax+ay$ \\
|
||||
& $(a+b)x~$&$~ax+bx$ \\
|
||||
Compatibility$^*$ & $(ab)x~$&$~a(bx)$ \\
|
||||
Identity & $a+0~$&$~a$
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
\vspace{2mm}
|
||||
|
||||
$^*$ Remember that $a, b \in \mathbb{F}$ and $x \in V$. Thus, $(ab)$ is multiplication in $\mathbb{F}$ and $(bx)$ is scalar multiplication in $V$. Compatibility is \textit{not} associativity. \\
|
||||
|
||||
Likewise, the addition you see in the distributive property of multiplication is field addition, not vector addition.
|
||||
|
||||
\vspace{6mm}
|
||||
|
||||
Usually, the word \textit{vector} refers to an element of $\mathbb{R}^n$. As you might expect $\mathbb{R}^n$ is a vector space over the field $\mathbb{R}$ under our usual vector operations.
|
||||
|
||||
Here's a quick review of these operations:
|
||||
\begin{itemize}
|
||||
\item Scalar multiplication is done elementwise: $3 \times [a, b, c] = [3a, 3b, 3c]$.
|
||||
\item Vector addition is similar: $[a, b, c] + [1, 2, 3] = [a+1,~b+2,~c+3]$.
|
||||
\item Vector addition is not valid for vectors of different sizes.
|
||||
\end{itemize}
|
||||
|
||||
\problem{}
|
||||
Verify that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$ under these operations.
|
||||
|
||||
\vfill
|
||||
|
||||
\pagebreak
|
||||
|
||||
We can also define an \textit{inner product} or \textit{vector product} that takes two elements of $V$ and produces another. \\
|
||||
|
||||
When we work in $\mathbb{R}^n$, we usually use the dot product as our vector product. It is defined as follows: \\
|
||||
|
||||
\definition{Dot Product}
|
||||
Given two vectors $a, b \in \mathbb{R}^n$, the \textit{dot product} of $a$ and $b$ (written $a \cdot b$ or $\langle a, b \rangle$) is $\sum_1^n a_ib_i$.
|
||||
|
||||
\vspace{2mm}
|
||||
|
||||
For example, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$,
|
||||
$$
|
||||
\langle a, b \rangle = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32
|
||||
$$
|
||||
As you may expect, the dot product $\langle a, b \rangle$ is valid iff $a$ and $b$ are the same size.
|
||||
|
||||
|
||||
|
||||
\problem{}
|
||||
Show that the dot product is commutative.
|
||||
|
||||
\vfill
|
||||
|
||||
\problem{}
|
||||
Show that the dot product is positive-definite. \\
|
||||
This means that $\langle a, a \rangle > 0$ unless $a = 0$.
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user