feat: 1522 lec 4,5,6
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@ -116,6 +116,7 @@ Examples of NON row echlon form:
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\subsection{Reduced Row-Echelon Form}
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\begin{defn}[Reduced Row-Echelon Form]
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\label{def:rref}
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Suppose an augmented matrix is in row-echelon form. It is in \textbf{reduced row-echelon form} if
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\begin{itemize}
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\item Leading entry of every nonzero row is 1
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\subsection{Homogeneous Linear System}
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\begin{defn}[Homogeneous Linear Equation \& System]\ where
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\begin{itemize}
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\item Homogeneous Linear Equation: $a_1x_1 + a_2x_2 + ... + a_nx_n = 0 \iff x_1 = 0, x_2 = 0,... , x_n = 0$
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\item Homogeneous Linear Equation: $\begin{cases}
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a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = 0 \\
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a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = 0 \\
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\vdots \\
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a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = 0 \\
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\end{cases}$
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\item This is the trivial solution of a homogeneous linear system.
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\end{itemize}
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You can use this to solve problems like Find the equation $ax^2 + by^2 + cz^2 = d$, in the $xyz$ plane which contains the points $(1, 1, -1), (1, 3, 3), (-2, 0, 2)$.
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\begin{itemize}
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\item Solve by first converting to Augmented Matrix, where the last column is all 0. During working steps, this column can be omitted.
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\item With the \hyperref[def:rref]{RREF}, you can set $d$ as $t$ and get values for $a, b, c$ in terms of $t$.
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\item sub in $t$ into the original equation and factorize $t$ out from both sides, for values where $t \neq 0$
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\end{itemize}
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\end{defn}
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\subsection{Matrix}
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\begin{defn}[Matrix]\ \\
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\begin{itemize}
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\item $\begin{pmatrix}
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a_{11} & a_{12} & ... & a_{1n} \\
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a_{21} & a_{22} & ... & a_{2n} \\
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\vdots \\
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a_{m1} & a_{m2} & ... & a_{mn}
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\end{pmatrix}$
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\item $m$ is no of rows, $n$ is no of columns
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\item size is $m \times n$
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\item $A = (a_{ij})_{m \times n}$
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\end{itemize}
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\end{defn}
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\subsection{Special Matrix}
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\begin{note}[Special Matrices]\ \\
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\begin{itemize}
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\item Row Matrix : $\begin{pmatrix} 2 & 1 & 0 \end{pmatrix}$
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\item Column Matrix
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\subitem $\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}$
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\item \textbf{Square Matrix}, $n \times n$ matrix / matrix of order $n$.
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\subitem Let $A = (a_{ij})$ be a square matrix of order $n$
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\item Diagonal of $A$ is $a_{11}, a_{22}, ..., a_{nn}$.
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\item \textbf{Diagonal Matrix} if Square Matrix and non-diagonal entries are zero
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\subitem Diagonals can be zero
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\subitem \textbf{Identity Matrix} is a special case of this
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\item \textbf{Square Matrix} if Diagonal Matrix and diagonal entries are all the same.
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\item \textbf{Identity Matrix} if Scalar Matrix and diagonal = 1
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\subitem $I_n$ is the identity matrix of order $n$.
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\item \textbf{Zero Matrix} if all entries are 0.
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\subitem Can denote by either $\overrightarrow{0}, 0$
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\item Square matrix is \textbf{symmetric} if symmetric wrt diagonal
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\subitem $A = (a_{ij})_{n \times n}$ is symmetric $\iff a_{ij} = a_{ji},\ \forall i, j$
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\item \textbf{Upper Triangular} if all entries \textbf{below} diagonal are zero.
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\subitem $A = (a_{ij})_{n \times n}$ is upper triangular $\iff a_{ij} = 0 \text{ if } i > j$
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\item \textbf{Lower Triangular} if all entries \textbf{above} diagonal are zero.
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\subitem $A = (a_{ij})_{n \times n}$ is lower triangular $\iff a_{ij} = 0 \text{ if } i < j$
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\subitem if Matrix is both Lower and Upper triangular, its a Diagonal Matrix.
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\end{itemize}
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\end{note}
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\subsection{Matrix Operations}
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\begin{defn}[Matrix Operations]\ \\
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Let $A = (a_{ij})_{m \times n}, B = (b_{ij})_{m \times n}$
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\begin{itemize}
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\item Equality: $B = (b_{ij})_{p \times q}$, $A = B \iff m = p \ \& \ n = q \ \& \ a_{ij} = b_{ij} \forall i,j$
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\item Addition: $A + B = (a_{ij} + b_{ij})_{m \times n}$
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\item Subtraction: $A - B = (a_{ij} - b_{ij})_{m \times n}$
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\item Scalar Mult: $cA = (ca_{ij})_{m \times n}$
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\end{itemize}
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\end{defn}
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\begin{defn}[Matrix Multiplication] \ \\
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Let $A = (a_{ij})_{m \times p}, B = (b_{ij})_{p \times n}$
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\begin{itemize}
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\item $AB$ is the $m \times n$ matrix s.t. $(i,j)$ entry is $$a_{i1}b_{1j} + a_{i2}b_{2j} + ... + a_{ip}b_{pj} = \sum^p_{k=1}a_{ik}b_{kj}$$
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\item No of columns in $A$ = No of rows in $B$.
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\item Matrix multiplication is \textbf{NOT commutative}
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\end{itemize}
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\end{defn}
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\begin{theorem}[Matrix Properties]\ \\
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Let $A, B, C$ be $m \times p, p \times q, q \times n$ matrices
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\begin{itemize}
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\item Associative Law: $A(BC) = (AB)C$
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\item Distributive Law: $A(B_1 + B_2) = AB_1 + AB_2$
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\item Distributive Law: $(B_1 + B_2)A = B_1A + B_2A$
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\item $c(AB) = (cA)B = A(cB)$
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\item $A\textbf{0}_{p \times n} = \textbf{0}_{m \times n}$
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\item $A\textbf{I}_{n} = \textbf{I}_{n}A = A$
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\end{itemize}
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\end{theorem}
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\begin{defn}[Powers of Square Matricss]\ \\
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Let $A$ be a $m \times n$.
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$AA$ is well defined $\iff m = n \iff A$ is square.
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\textbf{Definition.} Let $A$ be square matrix of order $n$. Then Powers of a are
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$$
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A^k = \begin{cases}
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I_n & \text{if } k = 0 \\
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AA...A & \text{if } k \geq 1.
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\end{cases}
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$$
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\textbf{Properties.}
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\begin{itemize}
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\item $A^mA^n = A^{m+n}, (A^m)^n = A^{mn}$
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\item $(AB)^2 = (AB)(AB) \neq A^2B^2 = (AA)(BB)$
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\end{itemize}
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\end{defn}
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Matrix Multiplication Example:
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\begin{itemize}
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\item Let $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \\ 2 & 3 \\ -1 & -2 \end{pmatrix}$
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\item Let $a_1 = \begin{pmatrix}1 & 2 & 3 \end{pmatrix}, a_2 = \begin{pmatrix}4 & 5 & 6 \end{pmatrix}$
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\item $AB = \begin{pmatrix} a_1 & a_2 \end{pmatrix}B = \begin{pmatrix} a_1B \\ a_2B \end{pmatrix}$.
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\item $\begin{pmatrix}
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\begin{pmatrix}1 & 2 & 3 \end{pmatrix} & \begin{pmatrix} 1 & 1 \\ 2 & 3 \\ -1 & -2 \end{pmatrix} \\
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\begin{pmatrix}4 & 5 & 6 \end{pmatrix} & \begin{pmatrix} 1 & 1 \\ 2 & 3 \\ -1 & -2 \end{pmatrix}
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\end{pmatrix} = \begin{pmatrix}
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\begin{pmatrix}2 & 1\end{pmatrix} \\
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\begin{pmatrix}8 & 7\end{pmatrix} \\
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\end{pmatrix}
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$
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\end{itemize}
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\begin{note}[Representation of Linear System] \ \\
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\begin{itemize}
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\item $\begin{cases}
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a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n & = b_1 \\
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a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n & = b_2 \\
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\vdots & \vdots \\
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a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n & = b_m \\
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\end{cases}$
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\item A = $\begin{pmatrix}
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a_{11} & a_{12} & ... & a_{1n} \\
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a_{21} & a_{22} & ... & a_{2n} \\
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\vdots & \vdots & & \vdots \\
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a_{m1} & a_{m2} & ... & a_{mn} \\
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\end{pmatrix}$, Coefficient Matrix, $A_{m\times n}$
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\item $x = \begin{pmatrix}
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x_{1} \\
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\vdots \\
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x_{n} \\
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\end{pmatrix}$, Variable Matrix, $x_{n \times 1}$
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\item $b = \begin{pmatrix}
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b_{1} \\
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\vdots \\
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b_{m} \\
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\end{pmatrix}$, Constant Matrix, $b_{m \times 1}$. Then $Ax = b$
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\item $A = (a_{ij})_{m\times n} $
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\item $m$ linear equations in $n$ variables, $x_1, ..., x_n$
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\item $a_{ij}$ are coefficients, $b_i$ are the constants
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\item Let $u = \begin{pmatrix} u_1 \\ \vdots \\ u_n \end{pmatrix}$.
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\subitem $x_1 = u_1, \hdots, x_n = u_n$ is a solution to the system
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\subitem $\iff Au = b \iff u$ is a solution to $Ax = b$
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\item Let $a_j$ denote the $j$th column of $A$. Then
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\subitem $b = Ax = x_1a_1 + ... + x_na_n = \sum^n_{j=1}x_ja_j$
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\end{itemize}
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\end{note}
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\begin{defn}[Transpose]\ \\
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\begin{itemize}
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\item Let $A = (a_{ij})_{m\times n}$
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\item The transpose of $A$ is $A^T = (a_{ji})_{n \times m}$
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\item $(A^T)^T = A$
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\item A is symmetric $\iff A = A^T$
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\item Let $B$ be $m \times n$, $(A+B)^T = A^T + B^T$
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\item Let $B$ be $n \times p$, $(AB)^T = B^TA^T$
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\end{itemize}
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\end{defn}
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\begin{defn}[Inverse]\ \\
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\begin{itemize}
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\item Let $A, B$ be matrices of same size
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\subitem $A + X = B \implies X = B - A = B + (-A)$
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\subitem $-A$ is the \textit{additive inverse} of $A$
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\item Let $A_{m\times n}, B_{m\times p}$ matrix.
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\subitem $AX = B \implies X = A^{-1}B$.
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\end{itemize}
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Let A be a \textbf{square matrix} of order $n$.
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\begin{itemize}
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\item If there exists a square matrix $B$ of order $N$ s.t. $AB = I_{n}$ and $BA = I_{n}$, then $A$ is \textbf{invertible} matrix and $B$ is inverse of $A$.
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\item If $A$ is not invertible, A is called singular.
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\item suppose $A$ is invertible with inverse $B$
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\item Let $C$ be any matrix having the same number of rows as $A$.
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$$\begin{aligned}
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AX = C &\implies B(AX) = BC \\
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&\implies (BA)X = BC \\
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&\implies X = BC.
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\end{aligned}$$
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\end{itemize}
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\end{defn}
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\begin{theorem}[Properties of Inversion]\ \\
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Let $A$ be a square matrix.
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\begin{itemize}
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\item Let $A$ be an invertible matrix, then its inverse is unique.
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\item Cancellation Law: Let $A$ be an invertible matrix
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\subitem $AB_1 = AB_2 \implies B_1 = B_2$
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\subitem $C_1A = C_2A \implies C_1 = C_2$
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\subitem $AB = 0 \implies B = 0, CA = 0 \implies C = 0$ ($A$ is invertible, A cannot be 0)
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\subitem This fails if $A$ is singular
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\item Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$
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\subitem $A$ is invertible $\iff ad - bc \neq 0$
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\subitem $A$ is invertible $A^{-1} = \dfrac{1}{ad - bc} \begin{pmatrix}d & -b \\ -c & a \end{pmatrix}$
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\end{itemize}
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Let $A$ and $B$ be invertible matrices of same order
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\begin{itemize}
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\item Let $c \neq 0$. Then $cA$ is invertible, $(cA^{-1} = \frac{1}{c}A^{-1}$
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\item $A^T$ is invertible, $(A^T)^{-1} = (A^{-1})^T$
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\item $AB$ is invertible, $(AB)^{-1} = (B^{-1}A^{-1})$
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\end{itemize}
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Let $A$ be an invertible matrix.
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\begin{itemize}
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\item $A^{-k} = (A^{-1})^k$
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\item $A^{m+n} = A^mA^n$
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\item $(A^m)^n = A^{mn}$
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\end{itemize}
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\end{theorem}
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\begin{defn}[Elementary Matrices] If it can be obtained from $I$ by performing single elementary row operation
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\begin{itemize}
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\item $cRi, c \neq 0: \begin{pmatrix}
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1 & 0 & 0 & 0 \\
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0 & 1 & 0 & 0 \\
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0 & 0 & c & 0 \\
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0 & 0 & 0 & 1
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\end{pmatrix}(cR_3)$
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\item $R_i \leftrightarrow R_j, i \neq j,: \begin{pmatrix}
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1 & 0 & 0 & 0 \\
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0 & 0 & 0 & 1 \\
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0 & 0 & 1 & 0 \\
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0 & 1 & 0 & 0
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\end{pmatrix}(R_2 \leftrightarrow R_4)$
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\item $R_i + cR_j, i \neq j,: \begin{pmatrix}
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1 & 0 & 0 & 0 \\
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0 & 1 & 0 & c \\
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0 & 0 & 1 & 0 \\
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0 & 0 & 0 & 1
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\end{pmatrix}(R_2 + cR_4)$
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\item Every elementary Matrix is invertible
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\end{itemize}
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\end{defn}
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$A = \begin{pmatrix}
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a_{11}&a_{12}&a_{13}\\
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a_{21}&a_{22}&a_{23}\\
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a_{31}&a_{32}&a_{33}\\
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a_{41}&a_{42}&a_{43}
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\end{pmatrix}$, $E = \begin{pmatrix}
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1&0&0&0\\
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0&1&0&0\\
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0&0&c&0\\
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0&0&0&1
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\end{pmatrix}(cR_3)$, $EA = \begin{pmatrix}
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a_{11}&a_{12}&a_{13}\\
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a_{21}&a_{22}&a_{23}\\
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ca_{31}&ca_{32}&ca_{33}\\
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a_{41}&a_{42}&a_{43}
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\end{pmatrix}$
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