diff --git a/Advanced/Linear Maps/main.tex b/Advanced/Linear Maps/main.tex index 6dfb23c..eda28dd 100755 --- a/Advanced/Linear Maps/main.tex +++ b/Advanced/Linear Maps/main.tex @@ -26,12 +26,9 @@ Prepared by Mark on \today \\ } - \section{Fields and Vector Spaces} - - \input{parts/0 fields} \input{parts/1 spaces} - \input{parts/2 linearity} + \input{parts/2 linear} \input{parts/3 matrices} @@ -40,7 +37,7 @@ \section{Bonus} \definition{} - Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space. + Show that $\mathbb{P}^n$ is a vector space. \vfill \problem{} diff --git a/Advanced/Linear Maps/parts/0 fields.tex b/Advanced/Linear Maps/parts/0 fields.tex index 4d42573..0df8b72 100644 --- a/Advanced/Linear Maps/parts/0 fields.tex +++ b/Advanced/Linear Maps/parts/0 fields.tex @@ -1,3 +1,5 @@ +\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 $ab$. \\ diff --git a/Advanced/Linear Maps/parts/1 spaces.tex b/Advanced/Linear Maps/parts/1 spaces.tex index c989c51..6331471 100644 --- a/Advanced/Linear Maps/parts/1 spaces.tex +++ b/Advanced/Linear Maps/parts/1 spaces.tex @@ -1,3 +1,5 @@ +\section{Spaces} + \definition{Vector Spaces} A \textit{space} over a field $\mathbb{F}$ consists of the following elements: \begin{itemize}[itemsep = 2mm] @@ -5,7 +7,7 @@ A \textit{space} over a field $\mathbb{F}$ consists of the following elements: \item An operation called \textit{vector addition}, denoted $+$ \\ Vector addition operates on two elements of $V$. \\ - \item An operation called \textit{scalar multilplication}, denoted $\times$ \\ + \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} diff --git a/Advanced/Linear Maps/parts/2 linearity.tex b/Advanced/Linear Maps/parts/2 linear.tex similarity index 89% rename from Advanced/Linear Maps/parts/2 linearity.tex rename to Advanced/Linear Maps/parts/2 linear.tex index e634f24..0720333 100644 --- a/Advanced/Linear Maps/parts/2 linearity.tex +++ b/Advanced/Linear Maps/parts/2 linear.tex @@ -1,4 +1,4 @@ -\section{Linearity} +\section{Linear Transformations} \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 $f: A \to B$. @@ -36,7 +36,8 @@ Is $\text{median}(v): \mathbb{R}^n \to \mathbb{R}$ a linear map on $\mathbb{R}^n \vfill \problem{} -Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? +Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? \\ +\hint{$\mathbb{P}^n$ is the set of all polynomials of degree $n$.} \vfill \pagebreak \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/3 matrices.tex b/Advanced/Linear Maps/parts/3 matrices.tex index 94dac64..fea0905 100644 --- a/Advanced/Linear Maps/parts/3 matrices.tex +++ b/Advanced/Linear Maps/parts/3 matrices.tex @@ -11,7 +11,8 @@ A = $$ The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix. -We can define the product of a matrix $A$ and a vector $v$ as follows: +\definition{} +We can define the product of a matrix $A$ and a vector $v$: $$ Av = @@ -62,9 +63,9 @@ Compute the following: $$ \begin{bmatrix} - 2 & 9 \\ - 7 & 5 \\ - 3 & 4 + 1 & 2 \\ + 3 & 4 \\ + 5 & 6 \end{bmatrix} \begin{bmatrix} 5 \\ 3 @@ -85,16 +86,16 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol $$ \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} $$ \end{center} @@ -111,9 +112,9 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol left delimiter={[}, right delimiter={]} ] (A) { - 2 & 9 \\ - 7 & 5 \\ + 1 & 2 \\ 3 & 4 \\ + 5 & 6 \\ }; \node[ @@ -134,21 +135,21 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol \node[ fit=(A-1-2)(A-1-2), inner xsep=8mm,inner ysep=0mm, - label=right:{$10 + 27 = 37$} + label=right:{$5 + 6 = 11$} ](Y) {}; \draw[->, gray] ([xshift=3mm]A-1-2.east) -- (Y); \node[ fit=(A-2-2)(A-2-2), inner xsep=8mm,inner ysep=0mm, - label=right:{$35 + 15 = 50$} + label=right:{$15 + 12 = 27$} ](H) {}; \draw[->, gray] ([xshift=3mm]A-2-2.east) -- (H); \node[ fit=(A-3-2)(A-3-2), inner xsep=8mm,inner ysep=0mm, - label=right:{$15 + 12 = 27$} + label=right:{$25 + 18 = 43$} ](N) {}; \draw[->, gray] ([xshift=3mm]A-3-2.east) -- (N); \end{tikzpicture} @@ -179,11 +180,6 @@ Show that the transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as $T(v) = Av$. \vfill -\pagebreak - -\problem{} -Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space. -\vfill \problem{} diff --git a/Advanced/Mock a Mockingbird/parts/02 kestrel.tex b/Advanced/Mock a Mockingbird/parts/02 kestrel.tex index 0e4f900..293b508 100644 --- a/Advanced/Mock a Mockingbird/parts/02 kestrel.tex +++ b/Advanced/Mock a Mockingbird/parts/02 kestrel.tex @@ -1,7 +1,7 @@ \section{The Curious Kestrel} \definition{} -Recall that a bird is \textit{egocenteric} if it is fond of itself. \\ +Recall that a bird is \textit{egocentric} if it is fond of itself. \\ A bird is \textit{hopelessly egocentric} if $Bx = B$ for all birds $x$. \definition{} @@ -33,7 +33,7 @@ $$ In other words, this means that for every bird $x$, the bird $Kx$ is fixated on $x$. \problem{} -Show that an egocenteric Kestrel is hopelessly egocentric. +Show that an egocentric Kestrel is hopelessly egocentric. \begin{solution} \begin{alltt} @@ -58,7 +58,7 @@ Given the Law of Composition and the Law of the Mockingbird, show that at least \end{helpbox} \begin{solution} - The final piece is a lemma we proved earler: \\ + The final piece is a lemma we proved earlier: \\ Any bird is fond of at least one bird \begin{alltt} @@ -115,7 +115,7 @@ Show that if $K$ is fond of $Kx$, $K$ is fond of $x$. An egocentric Kestrel must be extremely lonely. Why is this? \begin{solution} - If a Kestrel is egocenteric, it must be the only bird in the forest! + If a Kestrel is egocentric, it must be the only bird in the forest! \begin{alltt} \lineno{} \cmnt{Given} diff --git a/Misc/Warm-Ups/sin.tex b/Misc/Warm-Ups/sin.tex index d6fcf8a..c5e4ac4 100755 --- a/Misc/Warm-Ups/sin.tex +++ b/Misc/Warm-Ups/sin.tex @@ -19,7 +19,7 @@ \generic{Helpful identities:} This is not a complete list. In many cases, geometry is more helpful than algebra. \\ - Note that the first idenity is only valid if $\alpha \in [0, 90]$. + Note that the first identity is only valid if $\alpha \in [0, 90]$. \vspace{2mm} $\sin(\frac{\alpha}{2}) = \sqrt{\frac{1 - \cos(\alpha)}{2}}$ \\ @@ -32,7 +32,7 @@ \vspace{5mm} A good order to go in is 45, 30, 60, 15, 75, 36, 18, 3, 6, 72, 9, 1. \\ - You should be able to get all of these using only geometery and the identities above. + You should be able to get all of these using only geometry and the identities above. \end{solution} \end{document} \ No newline at end of file