diff --git a/Advanced/Linear Maps/main.tex b/Advanced/Linear Maps/main.tex index dc0c404..7a6c9c6 100755 --- a/Advanced/Linear Maps/main.tex +++ b/Advanced/Linear Maps/main.tex @@ -26,9 +26,6 @@ Prepared by Mark on \today \\ } - \section{Fields and Vector Spaces} - - \input{parts/0 fields} \input{parts/1 spaces} \input{parts/2 linearity} @@ -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 fe62669..9e05cad 100644 --- a/Advanced/Linear Maps/parts/3 matrices.tex +++ b/Advanced/Linear Maps/parts/3 matrices.tex @@ -17,8 +17,8 @@ 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.} +We can define the product of a matrix $A$ and a vector $v$: + $$ Av = \begin{bmatrix} @@ -34,7 +34,7 @@ Av = 4a + 5b + 6c \end{bmatrix} $$ -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 $v$: $$ Av = @@ -61,9 +61,9 @@ Compute the following: $$ \begin{bmatrix} - 2 & 9 \\ - 7 & 5 \\ - 3 & 4 + 1 & 2 \\ + 3 & 4 \\ + 5 & 6 \end{bmatrix} \begin{bmatrix} 5 \\ 3 @@ -84,16 +84,16 @@ 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} @@ -110,9 +110,9 @@ 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[ @@ -133,21 +133,21 @@ 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} @@ -187,11 +187,6 @@ Before you start, answer the following questions: 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{}