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