diff --git a/Advanced/Linear Maps/main.tex b/Advanced/Linear Maps/main.tex index eda28dd..914fb41 100755 --- a/Advanced/Linear Maps/main.tex +++ b/Advanced/Linear Maps/main.tex @@ -2,7 +2,8 @@ % use [solutions] flag to show solutions. \documentclass[ solutions, - singlenumbering + nowarning, + %singlenumbering ]{../../resources/ormc_handout} \usepackage{tikz} @@ -32,16 +33,44 @@ \input{parts/3 matrices} - - - \section{Bonus} + \section{Norms} \definition{} - Show that $\mathbb{P}^n$ is a vector space. + If $V$ is a vector space, a \textit{norm} in $V$ is a function $V \to \mathbb{R}^+$ that satisfies the following properties, \\ + Where $x, y \in V$ and $c \in F$: + + \begin{itemize} + \item Absolute Homogeneity: $||cx|| = |c|~||x||$ + \item Positive-Definite: $||x|| \geq 0$ with equality iff $x = 0$. + \item Triangle Inequalty: $||x+y|| \leq ||x|| + ||y||$ + \end{itemize} + + \problem{} + Show that the \textit{euclidian norm} defined by $||~[a, b]~|| = \sqrt{a^2 + b^2}$ is a norm on $\mathbb{R}^2$ + \vfill \problem{} - Show that the set of all linear maps is a vector space. + Show that in any vector space with an inner product, the \textit{induced norm} $||x|| = \sqrt{\langle x, x \rangle}$ is a norm. + + \vfill + + \problem{} + Show that every norm satisfies the reverse triangle inequality: + + $$ + ||x - y|| \geq |~||x|| - ||y||~| + $$ + + \vfill + + \problem{} + Prove the Cauchy-Schwartz inequality: + + $$ + ||\langle x, y \rangle|| = ||x||~||y|| + $$ + \vfill \end{document} \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/1 spaces.tex b/Advanced/Linear Maps/parts/1 spaces.tex index ca632e3..a636410 100644 --- a/Advanced/Linear Maps/parts/1 spaces.tex +++ b/Advanced/Linear Maps/parts/1 spaces.tex @@ -1,6 +1,6 @@ \section{Spaces} -\definition{Vector 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} @@ -46,7 +46,7 @@ In both tables, $x, y, z \in V$ and $a, b\in \mathbb{F}$. \begin{center} \begin{tabular}{l | r@{=}l } \hline - \multicolumn{3}{|c|}{Properties of vector multiplication} \\ + \multicolumn{3}{|c|}{Properties of scalar multiplication} \\ \hline Closure & \multicolumn{2}{c}{$ax \in V$} \\ Distributivity & $a(x+y)~$&$~ax+ay$ \\ @@ -59,39 +59,53 @@ In both tables, $x, y, z \in V$ and $a, b\in \mathbb{F}$. \vspace{2mm} -$^*$ Remember that $a, b \in \mathbb{F}$ and $x \in V$. Thus, $(ab)$ is scalar multiplication and $(bx)$ is vector multiplication. \\ -Compatibility is \textit{not} associativity, it is \say{compatibility of vector and scalar multiplication.} +$^*$ 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. \\ -\vfill -\pagebreak +Likewise, the addition you see in the distributive property of multiplication is field addition, not vector addition. -\definition{} -There is a good chance you are familiar with basic vector arithmetic. \\ -Here's a quick review: +\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. -\definition{} -We usually use the \textit{dot product} as our vector product. It is defined as follows. \\ -Given two vectors $a, b \in \mathbb{R}^n$, the dot product $a \cdot b$ is $\sum_1^n a_ib_i$. +\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} -In other words, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$, +For example, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$, $$ - a \cdot b = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32 + \langle a, b \rangle = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32 $$ -As you may expect, the dot product $ab$ is valid iff $a$ and $b$ are the same size. +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 satisfies the properties of a vector product listed above. \\ -Conclude that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$. +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 diff --git a/Advanced/Linear Maps/parts/3 matrices.tex b/Advanced/Linear Maps/parts/3 matrices.tex index fea0905..6dc2ce3 100644 --- a/Advanced/Linear Maps/parts/3 matrices.tex +++ b/Advanced/Linear Maps/parts/3 matrices.tex @@ -29,7 +29,7 @@ Av = 4a + 5b + 6c \end{bmatrix} $$ -Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$: +Note that each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$: $$ Av = @@ -191,10 +191,8 @@ Find a matrix that corresponds to $D$. \\ \vfill \pagebreak - \problem{} -Does \ref{thebigtheorem} hold in arbitrary vector spaces? \\ -Repeat \ref{prooffwd} and \ref{proofback} using only axioms, without assuming that we're working in $\mathbb{R}^n$. +Show that the set of all linear maps is a vector space. \vfill \pagebreak \ No newline at end of file