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Mark 2023-04-19 08:58:26 -07:00
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2
Advanced/Linear Algebra 101/main.tex
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@ -61,7 +61,7 @@
Prove the Cauchy-Schwartz inequality: Prove the Cauchy-Schwartz inequality:
$$ $$
||\langle x, y \rangle|| = ||x||~||y|| ||x \cdot y|| = ||x||~||y||
$$ $$
\vfill \vfill

2
Advanced/Linear Algebra 101/parts/1 vectors.tex
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@ -31,7 +31,7 @@ Can you develop geometric intuition for their sum and difference?
\begin{tikzpicture}[scale=1] \begin{tikzpicture}[scale=1]
\draw[->] \draw[->]
(0,0) coordinate (o) -- node[below left] {$(1, 2)$} (0,0) coordinate (o) -- node[below left] {$(2, -1)$}
(2, -1) coordinate (a) (2, -1) coordinate (a)
; ;

4
Advanced/Linear Algebra 101/parts/2 dotprod.tex
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@ -40,7 +40,7 @@ Show that the dot product is
\problem{} \problem{}
Say you have two vectors, $a$ and $b$. Show that $\langle a, b \rangle$ = $||a||~||b||\cos(\alpha)$ \\ Say you have two vectors, $a$ and $b$. Show that $a \cdot b$ = $||a||~||b||\cos(\alpha)$ \\
\hint{What is $c$ in terms of $a$ and $b$?} \hint{What is $c$ in terms of $a$ and $b$?}
\hint{The law of cosines is $a^2 + b^2 - 2ab\cos(\alpha) = c^2$} \hint{The law of cosines is $a^2 + b^2 - 2ab\cos(\alpha) = c^2$}
\hint{The length of $a$ is $||a||$} \hint{The length of $a$ is $||a||$}
@ -83,7 +83,7 @@ Say you have two vectors, $a$ and $b$. Show that $\langle a, b \rangle$ = $||a||
\vfill \vfill
\problem{} \problem{}
If $a$ and $b$ are perpendicular, what must $\langle a, b \rangle$ be? Is the converse true? If $a$ and $b$ are perpendicular, what must $a \cdot b$ be? Is the converse true?
\vfill \vfill

6
Advanced/Linear Algebra 101/parts/3 matrices.tex
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@ -76,7 +76,7 @@ $$
\pagebreak \pagebreak
\definition{} \definition{}
We also multiply a matrix by a matrix: We can also multiply a matrix by a matrix:
$$ $$
AB = AB =
@ -257,11 +257,11 @@ $
\vfill \vfill
\pagebreak \pagebreak
The \say{transpose} operator is often used to write column vectors compactly. \\ The \say{transpose} operator is often used to write column vectors in a compact way. \\
Vertical arrays don't look good in horizontal text. Vertical arrays don't look good in horizontal text.
\problem{} \problem{}
Consider the vectors $a = [1, 2, 3]^T$ and $b = [40, 50, 60]^T$ \\ Consider the vectors $a = [1, 4, 3]^T$ and $b = [9, 1, 4]^T$ \\
\begin{itemize} \begin{itemize}
\item Compute the dot product $ab$. \item Compute the dot product $ab$.
\item Can you redefine the dot product using matrix multiplication? \item Can you redefine the dot product using matrix multiplication?