From 998e802a99ca575e94067d131a48cdac2dd25b35 Mon Sep 17 00:00:00 2001 From: Yadunand Prem Date: Fri, 23 Feb 2024 00:23:46 +0800 Subject: [PATCH] feat: 1522 lec 4,5,6 --- ma1522/1522 Notes.pdf | Bin 74754 -> 98510 bytes ma1522/lec_02.tex | 1 + ma1522/lec_04.tex | 91 +++++++++++++++++++++++++++++++ ma1522/lec_05.tex | 121 ++++++++++++++++++++++++++++++++++++++++++ ma1522/lec_06.tex | 72 +++++++++++++++++++++++++ 5 files changed, 285 insertions(+) diff --git a/ma1522/1522 Notes.pdf b/ma1522/1522 Notes.pdf index decb26476c8d35e5aed1f97e105fcfdc52e76664..dc3bd317bcca04737232f72028c5ba53e9d216e4 100644 GIT binary patch delta 69609 zcmY(KV|1X+wyneN*tXNLZQHi3j-5BQ?R1=Ul8$ZLw%M_r`|Z8Yz4z>YbJVC&zp83J zHP@=Z5s0aPZ#Dj)U~J4uvlQe22O|?dKa8`BlbMknjK|tVmW;y&JCgs^2RirAkBNAe ze$4pGd_k38&0t-Pe3j>%r379!U%p`|8hflFjVPJ`;KI&eRuS3T+R>IC!MG9&6UQ>j_Ys2m{!9h`=If$_%@uIUNb&o zZs#bkE1Tee>GOq46vA**7I=JJpA6z3tln%5_uVskq%|s=b{(faEQh zKMsLsvdZT77fp7!I;vD4^ZDEPzRJ#z&bDm=Jz~Vj&$;VM7n!YL8oEPA*O-bH6dHKV 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row-echelon form. It is in \textbf{reduced row-echelon form} if \begin{itemize} \item Leading entry of every nonzero row is 1 diff --git a/ma1522/lec_04.tex b/ma1522/lec_04.tex index e69de29..5187958 100644 --- a/ma1522/lec_04.tex +++ b/ma1522/lec_04.tex @@ -0,0 +1,91 @@ +\subsection{Homogeneous Linear System} + +\begin{defn}[Homogeneous Linear Equation \& System]\ where + \begin{itemize} + \item Homogeneous Linear Equation: $a_1x_1 + a_2x_2 + ... + a_nx_n = 0 \iff x_1 = 0, x_2 = 0,... , x_n = 0$ + \item Homogeneous Linear Equation: $\begin{cases} + a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = 0 \\ + a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = 0 \\ + \vdots \\ + a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = 0 \\ + \end{cases}$ + + \item This is the trivial solution of a homogeneous linear system. + \end{itemize} + + 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 $(1, 1, -1), (1, 3, 3), (-2, 0, 2)$. + + \begin{itemize} + \item Solve by first converting to Augmented Matrix, where the last column is all 0. During working steps, this column can be omitted. + \item With the \hyperref[def:rref]{RREF}, you can set $d$ as $t$ and get values for $a, b, c$ in terms of $t$. + \item sub in $t$ into the original equation and factorize $t$ out from both sides, for values where $t \neq 0$ + \end{itemize} + +\end{defn} + +\subsection{Matrix} + +\begin{defn}[Matrix]\ \\ + \begin{itemize} + \item $\begin{pmatrix} + a_{11} & a_{12} & ... & a_{1n} \\ + a_{21} & a_{22} & ... & a_{2n} \\ + \vdots \\ + a_{m1} & a_{m2} & ... & a_{mn} + \end{pmatrix}$ + \item $m$ is no of rows, $n$ is no of columns + \item size is $m \times n$ + \item $A = (a_{ij})_{m \times n}$ + + \end{itemize} +\end{defn} + +\subsection{Special Matrix} + +\begin{note}[Special Matrices]\ \\ + \begin{itemize} + \item Row Matrix : $\begin{pmatrix} 2 & 1 & 0 \end{pmatrix}$ + \item Column Matrix + \subitem $\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}$ + \item \textbf{Square Matrix}, $n \times n$ matrix / matrix of order $n$. + \subitem Let $A = (a_{ij})$ be a square matrix of order $n$ + \item Diagonal of $A$ is $a_{11}, a_{22}, ..., a_{nn}$. + \item \textbf{Diagonal Matrix} if Square Matrix and non-diagonal entries are zero + \subitem Diagonals can be zero + \subitem \textbf{Identity Matrix} is a special case of this + \item \textbf{Square Matrix} if Diagonal Matrix and diagonal entries are all the same. + \item \textbf{Identity Matrix} if Scalar Matrix and diagonal = 1 + \subitem $I_n$ is the identity matrix of order $n$. + \item \textbf{Zero Matrix} if all entries are 0. + \subitem Can denote by either $\overrightarrow{0}, 0$ + \item Square matrix is \textbf{symmetric} if symmetric wrt diagonal + \subitem $A = (a_{ij})_{n \times n}$ is symmetric $\iff a_{ij} = a_{ji},\ \forall i, j$ + \item \textbf{Upper Triangular} if all entries \textbf{below} diagonal are zero. + \subitem $A = (a_{ij})_{n \times n}$ is upper triangular $\iff a_{ij} = 0 \text{ if } i > j$ + \item \textbf{Lower Triangular} if all entries \textbf{above} diagonal are zero. + \subitem $A = (a_{ij})_{n \times n}$ is lower triangular $\iff a_{ij} = 0 \text{ if } i < j$ + \subitem if Matrix is both Lower and Upper triangular, its a Diagonal Matrix. + \end{itemize} +\end{note} + +\subsection{Matrix Operations} + +\begin{defn}[Matrix Operations]\ \\ + Let $A = (a_{ij})_{m \times n}, B = (b_{ij})_{m \times n}$ + \begin{itemize} + \item Equality: $B = (b_{ij})_{p \times q}$, $A = B \iff m = p \ \& \ n = q \ \& \ a_{ij} = b_{ij} \forall i,j$ + \item Addition: $A + B = (a_{ij} + b_{ij})_{m \times n}$ + \item Subtraction: $A - B = (a_{ij} - b_{ij})_{m \times n}$ + \item Scalar Mult: $cA = (ca_{ij})_{m \times n}$ + \end{itemize} +\end{defn} + +\begin{defn}[Matrix Multiplication] \ \\ + Let $A = (a_{ij})_{m \times p}, B = (b_{ij})_{p \times n}$ + \begin{itemize} + \item $AB$ is the $m \times n$ matrix s.t. $(i,j)$ entry is $$a_{i1}b_{1j} + a_{i2}b_{2j} + ... + a_{ip}b_{pj} = \sum^p_{k=1}a_{ik}b_{kj}$$ + \item No of columns in $A$ = No of rows in $B$. + \item Matrix multiplication is \textbf{NOT commutative} + \end{itemize} +\end{defn} + diff --git a/ma1522/lec_05.tex b/ma1522/lec_05.tex index e69de29..8cb9728 100644 --- a/ma1522/lec_05.tex +++ b/ma1522/lec_05.tex @@ -0,0 +1,121 @@ +\begin{theorem}[Matrix Properties]\ \\ + Let $A, B, C$ be $m \times p, p \times q, q \times n$ matrices + \begin{itemize} + \item Associative Law: $A(BC) = (AB)C$ + \item Distributive Law: $A(B_1 + B_2) = AB_1 + AB_2$ + \item Distributive Law: $(B_1 + B_2)A = B_1A + B_2A$ + \item $c(AB) = (cA)B = A(cB)$ + \item $A\textbf{0}_{p \times n} = \textbf{0}_{m \times n}$ + \item $A\textbf{I}_{n} = \textbf{I}_{n}A = A$ + \end{itemize} +\end{theorem} + + +\begin{defn}[Powers of Square Matricss]\ \\ + Let $A$ be a $m \times n$. + + $AA$ is well defined $\iff m = n \iff A$ is square. + + \textbf{Definition.} Let $A$ be square matrix of order $n$. Then Powers of a are + $$ + A^k = \begin{cases} + I_n & \text{if } k = 0 \\ + AA...A & \text{if } k \geq 1. + \end{cases} + $$ + + \textbf{Properties.} + \begin{itemize} + \item $A^mA^n = A^{m+n}, (A^m)^n = A^{mn}$ + \item $(AB)^2 = (AB)(AB) \neq A^2B^2 = (AA)(BB)$ + \end{itemize} +\end{defn} + +Matrix Multiplication Example: + +\begin{itemize} + \item Let $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \\ 2 & 3 \\ -1 & -2 \end{pmatrix}$ + \item Let $a_1 = \begin{pmatrix}1 & 2 & 3 \end{pmatrix}, a_2 = \begin{pmatrix}4 & 5 & 6 \end{pmatrix}$ + \item $AB = \begin{pmatrix} a_1 & a_2 \end{pmatrix}B = \begin{pmatrix} a_1B \\ a_2B \end{pmatrix}$. + \item $\begin{pmatrix} + \begin{pmatrix}1 & 2 & 3 \end{pmatrix} & \begin{pmatrix} 1 & 1 \\ 2 & 3 \\ -1 & -2 \end{pmatrix} \\ + \begin{pmatrix}4 & 5 & 6 \end{pmatrix} & \begin{pmatrix} 1 & 1 \\ 2 & 3 \\ -1 & -2 \end{pmatrix} + \end{pmatrix} = \begin{pmatrix} + \begin{pmatrix}2 & 1\end{pmatrix} \\ + \begin{pmatrix}8 & 7\end{pmatrix} \\ + \end{pmatrix} + $ +\end{itemize} + +\begin{note}[Representation of Linear System] \ \\ + \begin{itemize} + \item $\begin{cases} + a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n & = b_1 \\ + a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n & = b_2 \\ + \vdots & \vdots \\ + a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n & = b_m \\ + \end{cases}$ + + \item A = $\begin{pmatrix} + a_{11} & a_{12} & ... & a_{1n} \\ + a_{21} & a_{22} & ... & a_{2n} \\ + \vdots & \vdots & & \vdots \\ + a_{m1} & a_{m2} & ... & a_{mn} \\ + \end{pmatrix}$, Coefficient Matrix, $A_{m\times n}$ + \item $x = \begin{pmatrix} + x_{1} \\ + \vdots \\ + x_{n} \\ + \end{pmatrix}$, Variable Matrix, $x_{n \times 1}$ + \item $b = \begin{pmatrix} + b_{1} \\ + \vdots \\ + b_{m} \\ + \end{pmatrix}$, Constant Matrix, $b_{m \times 1}$. Then $Ax = b$ + \item $A = (a_{ij})_{m\times n} $ + \item $m$ linear equations in $n$ variables, $x_1, ..., x_n$ + \item $a_{ij}$ are coefficients, $b_i$ are the constants + \item Let $u = \begin{pmatrix} u_1 \\ \vdots \\ u_n \end{pmatrix}$. + \subitem $x_1 = u_1, \hdots, x_n = u_n$ is a solution to the system + \subitem $\iff Au = b \iff u$ is a solution to $Ax = b$ + \item Let $a_j$ denote the $j$th column of $A$. Then + \subitem $b = Ax = x_1a_1 + ... + x_na_n = \sum^n_{j=1}x_ja_j$ + \end{itemize} +\end{note} + +\begin{defn}[Transpose]\ \\ + \begin{itemize} + \item Let $A = (a_{ij})_{m\times n}$ + \item The transpose of $A$ is $A^T = (a_{ji})_{n \times m}$ + \item $(A^T)^T = A$ + \item A is symmetric $\iff A = A^T$ + \item Let $B$ be $m \times n$, $(A+B)^T = A^T + B^T$ + \item Let $B$ be $n \times p$, $(AB)^T = B^TA^T$ + \end{itemize} +\end{defn} + +\begin{defn}[Inverse]\ \\ + \begin{itemize} + \item Let $A, B$ be matrices of same size + \subitem $A + X = B \implies X = B - A = B + (-A)$ + \subitem $-A$ is the \textit{additive inverse} of $A$ + \item Let $A_{m\times n}, B_{m\times p}$ matrix. + \subitem $AX = B \implies X = A^{-1}B$. + \end{itemize} + + + Let A be a \textbf{square matrix} of order $n$. + \begin{itemize} + \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 $A$. + \item If $A$ is not invertible, A is called singular. + \item suppose $A$ is invertible with inverse $B$ + \item Let $C$ be any matrix having the same number of rows as $A$. + $$\begin{aligned} + AX = C &\implies B(AX) = BC \\ + &\implies (BA)X = BC \\ + &\implies X = BC. + \end{aligned}$$ + \end{itemize} + + +\end{defn} diff --git a/ma1522/lec_06.tex b/ma1522/lec_06.tex index e69de29..00e3c26 100644 --- a/ma1522/lec_06.tex +++ b/ma1522/lec_06.tex @@ -0,0 +1,72 @@ +\begin{theorem}[Properties of Inversion]\ \\ + Let $A$ be a square matrix. + \begin{itemize} + \item Let $A$ be an invertible matrix, then its inverse is unique. + \item Cancellation Law: Let $A$ be an invertible matrix + \subitem $AB_1 = AB_2 \implies B_1 = B_2$ + \subitem $C_1A = C_2A \implies C_1 = C_2$ + \subitem $AB = 0 \implies B = 0, CA = 0 \implies C = 0$ ($A$ is invertible, A cannot be 0) + \subitem This fails if $A$ is singular + \item Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ + \subitem $A$ is invertible $\iff ad - bc \neq 0$ + \subitem $A$ is invertible $A^{-1} = \dfrac{1}{ad - bc} \begin{pmatrix}d & -b \\ -c & a \end{pmatrix}$ + + \end{itemize} + Let $A$ and $B$ be invertible matrices of same order + \begin{itemize} + \item Let $c \neq 0$. Then $cA$ is invertible, $(cA^{-1} = \frac{1}{c}A^{-1}$ + \item $A^T$ is invertible, $(A^T)^{-1} = (A^{-1})^T$ + \item $AB$ is invertible, $(AB)^{-1} = (B^{-1}A^{-1})$ + \end{itemize} + + Let $A$ be an invertible matrix. + + \begin{itemize} + \item $A^{-k} = (A^{-1})^k$ + \item $A^{m+n} = A^mA^n$ + \item $(A^m)^n = A^{mn}$ + \end{itemize} + + +\end{theorem} + +\begin{defn}[Elementary Matrices] If it can be obtained from $I$ by performing single elementary row operation + \begin{itemize} + \item $cRi, c \neq 0: \begin{pmatrix} + 1 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & c & 0 \\ + 0 & 0 & 0 & 1 + \end{pmatrix}(cR_3)$ + \item $R_i \leftrightarrow R_j, i \neq j,: \begin{pmatrix} + 1 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 1 \\ + 0 & 0 & 1 & 0 \\ + 0 & 1 & 0 & 0 + \end{pmatrix}(R_2 \leftrightarrow R_4)$ + \item $R_i + cR_j, i \neq j,: \begin{pmatrix} + 1 & 0 & 0 & 0 \\ + 0 & 1 & 0 & c \\ + 0 & 0 & 1 & 0 \\ + 0 & 0 & 0 & 1 + \end{pmatrix}(R_2 + cR_4)$ + \item Every elementary Matrix is invertible + \end{itemize} +\end{defn} + +$A = \begin{pmatrix} + a_{11}&a_{12}&a_{13}\\ + a_{21}&a_{22}&a_{23}\\ + a_{31}&a_{32}&a_{33}\\ + a_{41}&a_{42}&a_{43} +\end{pmatrix}$, $E = \begin{pmatrix} + 1&0&0&0\\ + 0&1&0&0\\ + 0&0&c&0\\ + 0&0&0&1 +\end{pmatrix}(cR_3)$, $EA = \begin{pmatrix} + a_{11}&a_{12}&a_{13}\\ + a_{21}&a_{22}&a_{23}\\ + ca_{31}&ca_{32}&ca_{33}\\ + a_{41}&a_{42}&a_{43} +\end{pmatrix}$