\input{decls.tex}
\title{Linear Algebra}
\begin{document}
\maketitle
\section*{Allowable Operations on a Linear System}
Solutions invariant.
\begin{itemize}
	\item Multiply an equation by a non-zero scalar
	\item Swap two equations
	\item Add a multiple of one equation to another
\end{itemize}
\subsection*{Example}
\begin{align*}
	    & \systeme{
		x - 2y + 2z = 6,
		-x + 3y + 4z = 2,
		2x + y - 2z = -2
	}                         \\\\
	E_2 & \implies E_2 + E_1  \\
	E_3 & \implies E_3 + E_1  \\
	    & \systeme{
		x - 2y + 2z = 6,
		y + 6z = 8,
		5y - 6z = -14
	}                         \\\\
	E_3 & \implies E_3 - 5E_2 \\
	    & \systeme{
		x - 2y + 2z = 6,
		y + 6z = 8,
		z = \frac{3}{2}
	}                         \\\\
	E_1 & \implies E_1 - 2E_3 \\
	E_2 & \implies E_2 - 6E_3 \\
	    & \systeme{
		x - 2y = 3,
		y = -1,
		z = \frac{3}{2}
	}                         \\\\
	E_1 & \implies E_1 + 2E_2 \\
	    & \systeme{
		x = 1,
		y = -1,
		z = \frac{3}{2}
	}                         \\\\
\end{align*}
\section*{As Matrices}
\begin{align*}
	\systeme{
		x + 2y = 1,
		2x - y = 3
	}
	\quad=\quad
	\begin{pmatrix}[cc|c]
		1 & 2  & 1 \\
		2 & -1 & 3
	\end{pmatrix}
	                               & \systeme{
		x - y + z = -2,
		2x + 3y + z = 7,
		x - 2y - z = -2
	} \quad=\quad \begin{pmatrix}[ccc|c]
		              1 & -1 & 1  & -2 \\
		              2 & 3  & 1  & 7  \\
		              1 & -2 & -1 & -2
	              \end{pmatrix}                    \\
	\grstep[R_3 - R_1]{R_2 - 2R_1} & \begin{pmatrix}[ccc|c]
		                                 1 & -1 & 1  & -2 \\
		                                 0 & 5  & -1 & 11 \\
		                                 0 & -1 & -2 & 0
	                                 \end{pmatrix} \\
	\grstep{5R_3 + R_2}            & \begin{pmatrix}[ccc|c]
		                                 1 & -1 & 1   & -2 \\
		                                 0 & 5  & -1  & 11 \\
		                                 0 & 0  & -11 & 11 \\
	                                 \end{pmatrix} \\
	\grstep{-11^{-1}R_3}           & \begin{pmatrix}[ccc|c]
		                                 1 & -1 & 1  & -2 \\
		                                 0 & 5  & -1 & 11 \\
		                                 0 & 0  & 1  & -1
	                                 \end{pmatrix} \\
	\grstep[R_1 - R_3]{R_2 + R_3}  & \begin{pmatrix}[ccc|c]
		                                 1 & -1 & 0 & -1 \\
		                                 0 & 5  & 0 & 10 \\
		                                 0 & 0  & 1 & -1
	                                 \end{pmatrix} \\&
	\grstep{5^{-1}R_2}             & \begin{pmatrix}[ccc|c]
		                                 1 & -1 & 0 & -1 \\
		                                 0 & 1  & 0 & 2  \\
		                                 0 & 0  & 1 & -1 \\
	                                 \end{pmatrix} \\
	\grstep{R_1 + R_2}             & \begin{pmatrix}[ccc|c]
		                                 1 & 0 & 0 & 1  \\
		                                 0 & 1 & 0 & 2  \\
		                                 0 & 0 & 1 & -1
	                                 \end{pmatrix} \\
	=                              & \quad
	\left\{
	\subalign{
	x                              & ~= ~1                 \\
	y                              & ~= ~2                 \\
	z                              & ~= ~-1
	}
	\right.
\end{align*}
\section*{Row-Echelon Form}
\begin{description}
	\item[Row-Echelon Form] The leading entry in each row is 1 and is further to the right than the previous row's leading entry,
		all 0 rows are at the end
	\item[Reduced Row-Echelon Form] every other entry in a column containing a leading 1 is 0
	\item[Theorem:] A matrix can be transformed to reduced row-echelon form using a finite number of allowable row operations
\end{description}
\subsection*{Example}
\begin{align*}
	                                & \systeme{3x_1 + 2x_2 = 1,
		x_1 - x_2 = 4,
	2x_1 + x_2 = 5} = \begin{pmatrix}[cc|c]
		                  3 & 2  & 1 \\
		                  1 & -1 & 4 \\
		                  2 & 1  & 5
	                  \end{pmatrix}                      \\
	\grstep{R_1\swap R_2}           & \begin{pmatrix}[cc|c]
		                                  1 & -1 & 4 \\
		                                  3 & 2  & 1 \\
		                                  2 & 1  & 5
	                                  \end{pmatrix}      \\
	\grstep[R_2 - 3R_1]{R_3 - 2R_1} & \begin{pmatrix}[cc|c]
		                                  1 & -1 & 4   \\
		                                  0 & 5  & -11 \\
		                                  0 & 3  & -3
	                                  \end{pmatrix}      \\
	\grstep{5^{-1}R_2}              & \begin{pmatrix}[cc|c]
		                                  1 & -1 & 4             \\
		                                  0 & 1  & \frac{-11}{5} \\
		                                  0 & 3  & -3
	                                  \end{pmatrix}    \\
	\grstep{R_3 - 2R_2}             & \begin{pmatrix}[cc|c]
		                                  1 & -1 & 4             \\
		                                  0 & 1  & \frac{-11}{5} \\
		                                  0 & 0  & \frac{18}{5}
	                                  \end{pmatrix}    \\
	=                               & \systeme{
		x_1 - x_2 = 4,
		x_2 = \frac{-11}{5},
		0x_1 + 0x_2 = \frac{18}{5}
	}
\end{align*}
\begin{align*}
	                                    & \begin{pmatrix}[cccc|c]
		                                      1  & -1 & 1  & 1 & 6 \\
		                                      -1 & 1  & -2 & 1 & 3 \\
		                                      2  & 0  & 1  & 4 & 1 \\
	                                      \end{pmatrix}                   \\
	\grstep[R_2 + R_1]{R_3 - 2R_1}      & \begin{pmatrix}[cccc|c]
		                                      1 & -1 & 1  & 1 & 6   \\
		                                      0 & 0  & -1 & 2 & 9   \\
		                                      0 & 2  & -1 & 2 & -11
	                                      \end{pmatrix}                   \\
	\grstep[R_2\swap R_3]{2^{-1}R_3}    & \begin{pmatrix}[cccc|c]
		                                      1 & -1 & 1           & 1 & 6             \\
		                                      0 & 1  & \frac{1}{2} & 1 & \frac{-11}{2} \\
		                                      0 & 0  & -1          & 2 & 9             \\
	                                      \end{pmatrix} \\
	\grstep[R_1 + R_3]{R_2 - 2^{-1}R_3} & \begin{pmatrix}[cccc|c]
		                                      1 & -1 & 0  & 3 & 15  \\
		                                      0 & 1  & 0  & 0 & -10 \\
		                                      0 & 0  & -1 & 2 & 9   \\
	                                      \end{pmatrix}                   \\
	\grstep[-R_3]{R_1 + R_2}            & \begin{pmatrix}[cccc|c]
		                                      1 & 0 & 0 & 3  & 15  \\
		                                      0 & 1 & 0 & 0  & -10 \\
		                                      0 & 0 & 1 & -2 & -9  \\
	                                      \end{pmatrix}                   \\
	=                                   & \systeme{
		x_1 + 3x_4 = 5,
		x_2 = -10,
		x_3 - 2x_4 = -9
	}                                                                              \\
	=                                   & \left\{\substack{
	x_1 = 5 - 3t                                                                   \\
	x_2 = -10                                                                      \\
		x_3 = -9 + 2t
	}\right.
\end{align*}
\section*{Determinants}
The determinant of a matrix is defined only for square matrices.
\[\det{A} \neq 0 \iff \exists \text{ a unique solution to the linear system represented by } A\]
Let
\[A = \begin{pmatrix}
		a_{11} & a_{12} & a_{1n} \\
		a_{21} & \ddots & \vdots \\
		a_{31} & \ldots & a_{3n} \\
	\end{pmatrix}
\]
\begin{description}
	\item[$i, j$ minor of $A$] an $n$x$n$ matrix constructed by removing the $i^\text{th}$ row and $j^\text{th}$ column of $A$ \\
		Denoted by $A_{ij}$
\end{description}
\begin{align*}
	                   & \det{A} \text{ where } n = 1. = a_{11}                                                                              \\
	                   & \det{A} = a_{11}\det{A_{11}} - a_{12}\det{A_{12}} + ... + (-1)^{n+1}a_{1n} \tag{Laplace expansion of the first row} \\
	                   & \qquad \text{or laplace expansion along other row or column}
	\text{For } n = 2: &                                                                                                                     \\
	                   & \det{A} = a_{11}\cdot a_{22} - a_{12}\cdot a_{21}
\end{align*}
\begin{description}
	\item[Upper Triangular] lower left triangle is 0 - $d_{ij} = 0 \quad \forall{i > j}$
	\item[Lower Triangular] upper right triangle is 0 - $d_{ij} = 0 \quad \forall{i < j}$
	\item[Diagonal] only values on the diagonal - $d_{ij} = 0 \quad \forall{i \neq j}$ \\
		$\det{A} = \prod^{N}_{i=0}~a_{ij} \forall~\text{ row-echelon }A$
\end{description}
\begin{itemize}
	\item Multiplying a row of a square matrix $A$ by $r$ multiplies $\det{A}$ by $r$
	\item Swapping two rows of a square matrix $A$ multiplies $\det{A}$ by $-1$
	\item Adding a multiple of a row does not effect the determinant
\end{itemize}
\section*{Transposition}
\begin{description}
	\item[$A^T$] $a^T_{ij} = a_{ji}~ \forall~i,j$
\end{description}
Note: $\det{A} = \det{A^T}~\forall~A$
\section*{Matrix Multiplication}
LHS has columns $=$ rows of RHS \\
It's the cartesian product
\[A\times B = (a_{i1}b_{j1} + a_{i2}b_{2j} + \ldots + a_{im}b_{mj})_{ij}\]
\begin{align*}
	\begin{pmatrix}[c|c|c]
		2    & 1 + 1 & 3 + 6    \\
		4(2) & 4 + 1 & 3(4) + 6 \\
		0    & 2     & 2(6)     \\
	\end{pmatrix} = \begin{pmatrix}
		                2 & 2 & 9  \\
		                8 & 5 & 18 \\
		                0 & 2 & 12
	                \end{pmatrix}
\end{align*}
\begin{align*}
	\begin{pmatrix}1 \\ 2 \\ 3 \end{pmatrix}\begin{pmatrix}1 & 2 & 3 & 4\end{pmatrix} + \begin{pmatrix}
		                                                                                    1 & 2  & 3  & 4  \\
		                                                                                    5 & 6  & 7  & 8  \\
		                                                                                    9 & 10 & 11 & 12 \\
	                                                                                    \end{pmatrix}
\end{align*}

\[A\vec{x} = \vec{b}\]
where $A$ is the coefficient matrix, $\vec{x}$ is the variables, and $\vec{b}$ is the values of the equations of a linear equation system.
\subsection*{Inverse Matrices}
The identity matrix exists as $I_n$ for size $n$.
\[AA^{-1} = I_n = A^{-1}A \quad \forall~\text{matrices }A \text{ of size } n\]
Assume that $A$ has two distinct inverses, $B$ and $C$.
\begin{align*}
	            & \text{matrix multiplication is associative} \\
	\therefore~ & C(AB) = (CA)B                               \\
	\therefore~ & C I_n = I_n B                               \\
	\therefore~ & C = B                                       \\
	            & \text{
		As $B = C$, while $B$ and $C$ are assumed to be distinct, matrices have no more than one unique inverse by contradiction
	}
\end{align*}
Matrices are invertible $\iff \det{A} \neq 0$
\[\det{AB} = \det{A}\det{B}\]
\[\therefore~ \det{A}\det{A^{-1}} = \det{I_n} = 1\]
\[\therefore~ \det{A} \neq 0 \]
\begin{align*}
	\begin{pmatrix}	a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\end{align*}
\subsubsection*{Computation thereof}
\[\det{A} = \sum_{k = 1}^{n}~a_{ik}(-1)^{i+j}\det{A_{ij}} \quad \text{ for any $i$}\]
\begin{description}
	\item[Matrix of Cofactors: $C$] determinants of minors \& signs of laplace expansion \\
		ie. $\sum A \odot C = \det{A}$
	\item[$\adj{A}$ Adjucate of $A$ =] $C^T$
\end{description}
\begin{align*}
	A    & = \begin{pmatrix}
		         1  & 0 & 1 \\
		         -1 & 1 & 2 \\
		         2  & 0 & 1
	         \end{pmatrix} \\
	C(A) & = \begin{pmatrix}
		         1  & 5  & -2 \\
		         0  & -1 & 0  \\
		         -1 & -3 & 1  \\
	         \end{pmatrix}
\end{align*}
$$ A^{-1} = \frac{\adj{A}}{\det{A}} $$
Gaussian elimination can also be used: augmented matrix with $I_n$ on the right,
reduce to reduced row-echelon. If the left is of the form $I_n$, the right is
the inverse. If there is a zero row, $\det{A} = 0$, and the $A$ has no inverse.
\section*{Linear Transformations}
\begin{align*}
	f: & ~ \R^n \to \R^m                                                                                    \\
	f  & (x_1, \cdots, x_n) = (f_1(x_1, \cdots, x_n), f_2(x_1, \cdots, x_n), \cdots, f_m(x_1, \cdots, x_n))
\end{align*}
$f$ is a linear transformation if \(\forall i.~f_i(x_1, \cdots, x_n)\) is a
linear polynomial in $x_1, \cdots, x_n$ with a zero constant term
\begin{align*}
	f(x_1,~ x_2)       & = (x_1 + x_2,~ 3x_1 - x_2,~ 10x_2) \tag{is a linear transformation} \\
	g(x_1,~ x_2,~ x_3) & = (x_1 x_2,~ x_3^2) \tag{not a linear transformation}               \\
	h(x_1,~ x_2)       & = (3x_1 + 4,~ 2x_2 - 4) \tag{not a linear transformation}           \\
\end{align*}
\[f: \R^n \to \R^m = \vec{x} \to A\vec{x} \]
\[\exists \text{ a matrix $A$ of dimension $n$x$m$ } \forall\text{ linear transforms } f \]
\[\forall \text{ matrices $A$ of dimension $n$x$m$ } \exists \text{ a linear transform $f$ of dimension $n$x$m$ such that } f(\vec{x}) = A\vec{x} \]
Function composition of linear translations is is just matrix multiplication:
\begin{align*}
	f(\vec{x})          & = A\vec{x}                  \\
	g(\vec{y})          & = B\vec{y}                  \\
	(f\cdot g)(\vec{x}) & = g(f(\vec{x})) = BA\vec{x}
\end{align*}
A function \(f: \R^n \to \R^m\) is a linear transformation iff:
\begin{enumerate}
	\item $f(\vec{x} + \vec{y}) = f(\vec{x}) + f(\vec{y}) \quad \forall~\vec{x},~\vec{y} \in \R^n $
	\item $f(r\vec{x}) = r\cdot f(\vec{x}) \quad \forall~\vec{x} \in \R^n, r \in \R $
\end{enumerate}
\subsection*{Building the matrix of a linear transform}
\[ f(\vec{x}) = f(x_1\vec{e}_1 + x_2\vec{e}_2) = f(x_1\vec{e}_1) + f(x_2\vec{e}_2) = x_1f(\vec{e}_1) + x_2f(\vec{e}_2) \]
\[ A = \begin{pmatrix} f(\vec{e}_1) & f(\vec{e}_2) \end{pmatrix} \]
\begin{align*}
	 & \vec{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}
	\\ & \vec{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}
	\\ & \vdots
	\\ & \forall \vec{x}.~ \vec{x} = \sum_{i}^{n}~\vec{e}_i x_i
\end{align*}
\subsection*{Composition}
\[ \paren{f \cdot g}\paren{\vec{x}} = f(g(\vec{x})) = AB\vec{x} \]
where: $f(\vec{x}) = A\vec{x}$, $g(\vec{x}) = B\vec{x}$
\subsection*{Geometry}
\begin{description}
	\item[rotation of $x$ by $\theta$ anticlockwise] \( = R_\theta = \begin{pmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{pmatrix} \)
	\item[reflection about a line at angle $\alpha$ from the $x$-axis] \( = T_\alpha = R_{\alpha}T_0R_{-\alpha}\) where \( T_0 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \)
	\item[scaling by $\lambda \in \R$] \( = S_\lambda = \lambda I_n\)
	\item[Skew by $\alpha$ in $x$ and $\gamma$ in $y$] \( \begin{pmatrix} \alpha & 0 \\ 0 & \gamma \end{pmatrix}\)
\end{description}
The image of the unit square under the linear transform $A$ is a parallelogram of $(0, 0)$, $(a_{11}, a_{21})$, $(a_{12}, a_{22})$, $(a_{11} + a_{12}, a_{21} + a_{22})$, with area $ \abs{\det{A}} $
\subsection*{Inversion}
Inversion of a linear transformation is equivalent to inversion of its representative matrix
\subsection*{Eigen\{values, vectors\}}
\[ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} a \\ 0 \end{pmatrix} = a\vec{e}_1\]
\[ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ b \end{pmatrix} = b\vec{e}_2\]
\[ T_\alpha \vec{x} = \vec{x} \text{ for $\vec{x}$ along the line of transformation }\]
\begin{description}
	\item[Eigenvector (of some transformation $f$)] A non-zero vector $\vec{x}$ such that $f(\vec{x}) = \lambda\vec{x}$ for some value $\lambda$
	\item[Eigenvalue] $\lambda$ as above
\end{description}
\[ \forall \text{ eigenvectors of $A$ } \vec{x}, c \in R, \neq 0 .~ c\vec{x} \text{ is an eigenvector with eigenvalue } \lambda\]

\[ \forall A: \text{$n$x$n$ matrix}.\quad P_A\paren{\lambda} = \det{\paren{A - \lambda I_n}} \tag{characteristic polynomial in $\lambda$}\]
Eigenvalues of $A$ are the solutions of $P_A\paren{\lambda} = 0$
\begin{align*}
	& A\vec{x} = \lambda\vec{x} & x \neq 0\\
	\iff & A\vec{x} - \lambda\vec{x} = 0 \\
	\iff & (A - \lambda I_n)\vec{x} = 0 \\
	\iff & \det{\paren{A - \lambda I_n}} = 0 \\
	& \quad \text{ or $\paren{A - \lambda I_n}$ is invertible and $x = 0$ }
\end{align*}
\[ P_{R\theta}(\lambda) = \frac{2\cos{\theta} \pm \sqrt{-4\lambda^2\sin^2{\theta}}}{2}\]
\[ R_\theta \text{ has eigenvalues }\iff \sin{\theta} = 0 \]
\end{document}