diff --git a/ma1522/1522 Notes.pdf b/ma1522/1522 Notes.pdf index cc71dee..b9cec55 100644 Binary files a/ma1522/1522 Notes.pdf and b/ma1522/1522 Notes.pdf differ diff --git a/ma1522/1522 Notes.tex b/ma1522/1522 Notes.tex index e13cf01..503d854 100644 --- a/ma1522/1522 Notes.tex +++ b/ma1522/1522 Notes.tex @@ -29,58 +29,19 @@ % ------------------------------------------------------------------------------ -\section{Lecture 1} +\section{Linear Systems} \hr -\input{lec_01.tex} +\input{ch_01.tex} \newpage -\section{Lecture 2} +\section{Matrices} \hr -\input{lec_02.tex} +\input{ch_02.tex} \newpage -\section{Lecture 3} + +\section{Vector Spaces} \hr -\input{lec_03.tex} -\newpage -\section{Lecture 4} -\hr -\input{lec_04.tex} -\newpage -\section{Lecture 5} -\hr -\input{lec_05.tex} -\newpage -\section{Lecture 6} -\hr -\input{lec_06.tex} -\newpage -\section{Lecture 7} -\hr -\input{lec_07.tex} -\newpage -\section{Lecture 8} -\hr -\input{lec_08.tex} -\newpage -\section{Lecture 9} -\hr -\input{lec_09.tex} -\newpage -\section{Lecture 10} -\hr -\input{lec_10.tex} -\newpage -\section{Lecture 11} -\hr -\input{lec_11.tex} -\newpage -\section{Lecture 12} -\hr -\input{lec_12.tex} -\newpage -\section{Lecture 13} -\hr -\input{lec_13.tex} +\input{ch_03.tex} \newpage \section{Reference} diff --git a/ma1522/lec_02.tex b/ma1522/ch_01.tex similarity index 54% rename from ma1522/lec_02.tex rename to ma1522/ch_01.tex index 3b32669..5a0fdc5 100644 --- a/ma1522/lec_02.tex +++ b/ma1522/ch_01.tex @@ -1,3 +1,179 @@ +\subsection{Linear Algebra} +\begin{itemize} + \item \textbf{Linear} The study of items/planes and objects which are flat + \item \textbf{Algebra} Objects are not as simple as numbers +\end{itemize} + +\subsection{Linear Systems \& Their Solutions} + +Points on a straight line are all the points $(x, y)$ on the $xy$ plane satisfying the linear eqn: $ax + by = c$, where $a, b > 0$ + +\subsubsection{Linear Equation} +Linear eqn in $n$ variables (unknowns) is an eqn in the form +$$ a_1x_1 + a_2x_2 + ... + a_nx_n = b$$ +where $a_1, a_2, ..., a_n, b$ are constants. + +\begin{note} +In a linear system, we don't assume that $a_1, a_2, ..., a_n$ are not all 0 +\begin{itemize} + \item If $a_1 = ... = a_n = 0$ but $b \neq 0$, it is \textbf{inconsistent} + + E.g. $0x_1 + 0x_2 = 1$ + + \item If $a_1 = ... = a_n = b = 0$, it is a \textbf{zero equation} + + E.g. $0x_1 + 0x_2 = 0$ + \item Linear equation which is not a zero equation is a \textbf{nonzero equation} + + E.g. $2x_1 - 3x_2 = 4$ + \item The following are not linear equations + \begin{itemize} + \item $xy = 2$ + \item $\sin\theta + \cos\phi = 0.2$ + \item $x_1^2 + x_2^2 + ... + x_n^2 = 1$ + \item $x = e^y$ + \end{itemize} +\end{itemize} +\end{note} + +In the $xyz$ space, linear equation $ax + by + cz = d$ where $a, b, c > 0$ represents a plane + +\subsubsection{Solutions to a Linear Equation} +Let $a_1x_1 + a_2x_2 + ... + a_nx_n = b$ be a linear eqn in n variables \\ +For real numbers $s_1+ s_2+ ... + s_n$, if $a_1s_1 + a_2s_2 + ... + a_ns_n = b$, then $x_1 = s_1, x_2 = s_2, x_n = s_n$ is a solution to the linear equation \\ +The set of all solutions is the \textbf{solution set}\\ +Expression that gives the entire solution set is the \textbf{general solution} + +\textbf{Zero Equation} is satified by any values of $x_1, x_2,... x_n$ + +General solution is given by $(x_1, x_2, ..., x_n) = (t_1, t_2, ..., t_n)$ + + +\subsubsection{Examples: Linear equation $4x-2y = 1$} +\begin{itemize} + \item x can take any arbitary value, say t + \item $x = t \Rightarrow y = 2t - \frac{1}{2}$ + \item General Solution: + $ + \begin{cases} + x = t & \text{t is a parameter}\\ + y = 2t - \frac{1}{2} + \end{cases} + $ + \item y can take any arbitary value, say s + \item $y = s \Rightarrow x = \frac{1}{2}s + \frac{1}{4}$ + \item General Solution: + $ + \begin{cases} + y = s & \text{s is a parameter}\\ + x = \frac{1}{2}s + \frac{1}{4} + \end{cases} + $ +\end{itemize} + +\subsubsection{Example: Linear equation $x_1 - 4x_2 + 7x_3 = 5$} + +\begin{itemize} + \item $x_2$ and $x_3$ can be chosen arbitarily, $s$ and $t$ + \item $x_1 = 5 + 4s -7t$ + \item General Solution: + $ + \begin{cases} + x_1 = 5 + 4s -7t \\ + x_2 = s & s, t \text{ are arbitrary parameters}\\ + x_3 = t \\ + \end{cases} + $ +\end{itemize} + + +\subsection{Linear System} +Linear System of m linear equations in n variables is +\begin{equation} +\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 \\ + a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_m \\ +\end{cases} +\end{equation} +where $a_{ij}, b$ are real constants and $a_{ij}$ is the coeff of $x_j$ in the $i$th equation + +\begin{note} Linear Systems + \begin{itemize} + \item If $a_{ij}$ and $b_i$ are zero, linear system is called a \textbf{zero system} + \item If $a_{ij}$ and $b_i$ is nonzero, linear system is called a \textbf{nonzero system} + \item If $x_1 = s_1, x_2 = s_2, ..., x_n = s_n$ is a solution to \textbf{every equation} in the system, then its a solution to the system + \item If every equation has a solution, there might not be a solution to the system + \item \textbf{Consistent} if it has at least 1 solution + \item \textbf{Inconsistent} if it has no solutions + \end{itemize} +\end{note} + + +\subsubsection{Example} +\begin{equation} +\begin{cases} +a_1x + b1_y = c_1 \\ +a_2x + b2_y = c_2 \\ +\end{cases} +\end{equation} + +where $a_1, b_1, a_2, b_2$ not all zero + +In $xy$ plane, each equation represents a straight line, $L_1, L_2$ + +\begin{itemize} + \item If $L_1, L_2$ are parallel, there is no solution + \item If $L_1, L_2$ are not parallel, there is 1 solution + \item If $L_1, L_2$ coinside(same line), there are infinitely many solution +\end{itemize} + +\begin{equation} +\begin{cases} +a_1x + b1_y + c_1z = d_1 \\ +a_2x + b2_y + c_2z = d_2 \\ +\end{cases} +\end{equation} + +where $a_1, b_1, c_1, a_2, b_2, c_2$ not all zero + +In $xyz$ space, each equation represents a plane, $P_1, P_2$ + +\begin{itemize} + \item If $P_1, P_2$ are parallel, there is no solution + \item If $P_1, P_2$ are not parallel, there is $\infty$ solutions (on the straight line intersection) + \item If $P_1, P_2$ coinside(same plane), there are infinitely many solutions + \item Same Plane $\Leftrightarrow a_1 : a_2 = b_1 : b_2 = c_1 : c_2 = d_1: d_2$ + \item Parallel Plane $\Leftrightarrow a_1 : a_2 = b_1 : b_2 = c_1 : c_2$ + \item Intersect Plane $\Leftrightarrow a_1 : a_2, b_1 : b_2, c_1 : c_2$ are not the same +\end{itemize} + +\subsection{Augmented Matrix} +$ + \begin{amatrix}{3} + a_{11} & a_{12} & a_{1n} & b_1 \\ + a_{21} & a_{12} & a_{2n} & b_2 \\ + a_{m1} & a_{m2} & a_{mn} & b_m \\ + \end{amatrix} +$ + +\subsection{Elementary Row Operations} +To solve a linear system we perform operations: +\begin{itemize} + \item Multiply equation by nonzero constant + \item Interchange 2 equations + \item add a constant multiple of an equation to another +\end{itemize} + +Likewise, for a augmented matrix, the operations are on the \textbf{rows} of the augmented matrix + +\begin{itemize} + \item Multiply row by nonzero constant + \item Interchange 2 rows + \item add a constant multiple of a row to another row +\end{itemize} + \subsection{Recap} Given the linear equation $a_1x_1 + a_2x_2 + ... + a_nx_n = b$ @@ -350,4 +526,88 @@ Given the augmented matrix is in row-echelon form. \end{enumerate} \end{note} +\subsection{Review} +\begin{align*} + I: & cR_i, c \neq 0 \\ + II: & R_i \Leftrightarrow R_j \\\ + III: & R_i \Rightarrow R_i + cR_j +\end{align*} + +Solving REF: +\begin{enumerate} + \item Set var -> non-pivot cols as params + \item Solve var -> pivot cols backwards + + \# of nonzero rows = \# pivot pts = \# of pivot cols +\end{enumerate} + +Gaussian Elimination +\begin{enumerate} + \item Given a matrix $A$, find left most non-zero \textbf{column}. If the leading number is NOT zero, use $II$ to swap rows. + \item Ensure the rest of the column is 0 (by subtracting the current row from tht other rows) + \item Cover the top row and continue for next rows +\end{enumerate} + +\subsection{Consistency} +\begin{defn}[Consistency]\ \\ + Suppose that $A$ is the Augmented Matrix of a linear system, and $R$ is a row-echelon form of $A$. + \begin{itemize} + \item When the system has no solution(inconsistent)? + \subitem There is a row in $R$ with the form $(0 0 ... 0 | \otimes)$ where $\otimes \neq 0$ + \subitem Or, the last column is a pivot column + \item When the system has exactly one solution? + \subitem Last column is non-pivot + \subitem All other columns are pivot columns + \item When the system has infinitely many solutions? + \subitem Last column is non-pivot + \subitem Some other columns are non-pivot columns. + \end{itemize} +\end{defn} + +\begin{note} Notations\ \\ + For elementary row operations + \begin{itemize} + \item Multiply $i$th row by (nonzero) const $k$: $kR_i$ + \item Interchange $i$th and $j$th rows: $R_i \leftrightarrow R_j$ + \item Add $K$ times $i$th row to $j$th row: $R_j + kR_i$ + \end{itemize} + \textbf{Note} + \begin{itemize} + \item $R_1 + R_2$ means "add 2nd row to the 1st row". + \item $R_2 + R_1$ means "add 1nd row to the 2st row". + \end{itemize} + + \textbf{Example} +$$ \begin{pmatrix} a \\ b \end{pmatrix} +\xrightarrow{R_1 + R_2} \begin{pmatrix} a + b \\ b \end{pmatrix} +\xrightarrow{R_2 + (-1)R_1} \begin{pmatrix} a + b \\ -a \end{pmatrix} +\xrightarrow{R_1 + R_2} \begin{pmatrix} b \\ -a \end{pmatrix} +\xrightarrow{(-1)R_2}\begin{pmatrix} b \\ a \end{pmatrix} + $$ +\end{note} + +\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} diff --git a/ma1522/ch_02.tex b/ma1522/ch_02.tex new file mode 100644 index 0000000..59e507b --- /dev/null +++ b/ma1522/ch_02.tex @@ -0,0 +1,446 @@ +\subsection{Introduction} + +\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} + +\subsubsection{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. + \label{def:ltm} + \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} + +\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} + +\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} + +$A = \begin{pmatrix} + a_{11}&a_{12}&a_{13}\\ + a_{21}&a_{22}&a_{23}\\ + a_{31}&a_{32}&a_{33}\\ + a_{41}&a_{42}&a_{43} +\end{pmatrix}$, $E = \begin{pmatrix} + 1&0&0&0\\ + 0&1&0&0\\ + 0&0&c&0\\ + 0&0&0&1 +\end{pmatrix}(cR_3)$, $EA = \begin{pmatrix} + a_{11}&a_{12}&a_{13}\\ + a_{21}&a_{22}&a_{23}\\ + ca_{31}&ca_{32}&ca_{33}\\ + a_{41}&a_{42}&a_{43} +\end{pmatrix}$ + +\begin{theorem} Main Theorem for Invertible Matrices \\ + Let $A$ be a square matrix. Then the following are equivalent + \begin{enumerate} + \item $A$ is an invertible matrix. + \item Linear System $Ax = b$ has a unique solution + \item Linear System $Ax = 0$ has only the trivial solution + \item RREF of $A$ is $I$ + \item A is the product of elementary matrices + \end{enumerate} +\end{theorem} + +\begin{theorem} Find Inverse + \begin{itemize} + \item Let $A$ be an invertible Matrix. + \item RREF of $(A | I)$ is $(I | A^{-1})$ + \end{itemize} + + How to identify if Square Matrix is invertible? + + \begin{itemize} + \item Square matrix is invertible + \subitem $\iff$ RREF is $I$ + \subitem $\iff$ All columns in its REF are pivot + \subitem $\iff$ All rows in REF are nonzero + \item Square matrix is singular + \subitem $\iff$ RREF is \textbf{NOT} $I$ + \subitem $\iff$ Some columns in its REF are non-pivot + \subitem $\iff$ Some rows in REF are zero. + \item $A$ and $B$ are square matrices such that $AB = I$ + \subitem then $A$ and $B$ are invertible + \end{itemize} +\end{theorem} + +\begin{defn}[LU Decomposition with Type 3 Operations]\ \\ + \begin{itemize} + \item Type 3 Operations: $(R_i + cR_j, i > j)$ + \item Let $A$ be a $m \times n$ matrix. Consider Gaussian Elimination $A \dashrightarrow R$ + \item Let $R \dashrightarrow A$ be the operations in reverse + \item Apply the same operations to $I_m \dashrightarrow L$. Then $A = LR$ + \item $L$ is a \hyperref[def:ltm]{lower triangular matrix} with 1 along diagonal + \item If $A$ is square matrix, $R = U$ + \end{itemize} + + Application: + \begin{itemize} + \item $A$ has LU decomposition $A = LU$, $Ax = b$ i.e., $LUx = b$ + \item Let $y = Ux$, then it is reduced to $Ly = b$ + \item $Ly = b$ can be solved with forward substitution. + \item $Ux = y$ is the REF of A. + \item $Ux = y$ can be solved using backward substitution. + + \end{itemize} +\end{defn} + +\begin{defn}[LU Decomposition with Type II Operations]\ \\ + \begin{itemize} + \item Type 2 Operations: $(R_i \leftrightarrow R_j)$, where 2 rows are swapped + \item $A \xrightarrow[]{E_1} \bullet \xrightarrow[]{E_2}\bullet \xrightarrow[E_3]{R_i \iff R_j}\bullet \xrightarrow[]{E_4}\bullet \xrightarrow[]{E_5} R$ + \item $A = E^{-1}_1E^{-1}_2E^{}_3E^{-1}_4E^{-1}_5R$ + \item $E_3A = (E_3E^{-1}_1E^{-1}_2E_3)E^{-1}_4E^{-1}_5R$ + \item $P = E_3, L = (E_3E^{-1}_1E^{-1}_2E_3)E^{-1}_4E^{-1}_5, R = U$, $PA = LU$ + + \end{itemize} +\end{defn} + +\begin{defn}[Column Operations]\ \\ + \begin{itemize} + \item Pre-multiplication of Elementary matrix $\iff$ Elementary row operation + \subitem $A \to B \iff B = E_1E_2...E_kA$ + \item Post-Multiplication of Elementary matrix $\iff$ Elementary Column Operation + \subitem $A \to B \iff B = AE_1E_2...E_k$ + \item If $E$ is obtained from $I_n$ by single elementary column operation, then + \subitem $I \xrightarrow[]{kC_i}E \iff I \xrightarrow[]{kR_i}E$ + \subitem $I \xrightarrow[]{C_i \leftrightarrow C_j}E \iff I \xrightarrow[]{R_i \leftrightarrow R_j}E$ + \subitem $I \xrightarrow[]{C_i + kC_j}E \iff I \xrightarrow[]{R_j + kR_i}E$ + \end{itemize} +\end{defn} + +\subsection{Determinants} + +\begin{defn}[Determinants of $2 \times 2$ Matrix]\ \\ + \begin{itemize} + \item Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ + \item $\det(A) = |A| = ad - bc$ + \end{itemize} + Solving Linear equations with determinants for $2 \times 2$ + \begin{itemize} + \item $x_1 = + \dfrac{\begin{vmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{vmatrix}} + {\begin{vmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}}$, $x_2 = + \dfrac{\begin{vmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{vmatrix}} + {\begin{vmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}}$ + \end{itemize} +\end{defn} + +\begin{defn}[Determinants]\ \\ + \begin{itemize} + \item Suppose $A$ is invertible, then there exists EROs such that + \item $A \xrightarrow{ero_1} A_1 \rightarrow ... \rightarrow A_{k-1} \xrightarrow{ero_k}A_k = I$ + \item Then $\det(A)$ can be evaluated backwards. + \subitem E.g. $A \xrightarrow{R_1 \leftrightarrow R_3} \bullet \xrightarrow{3R_2} \bullet \xrightarrow{R_2 + 2R_4} I \implies det(A) = 1 \to 1 \to \frac{1}{3} \to -\frac{1}{3}$ + \item Let $M_{ij}$ be submatrix where the $i$th row and $j$th column are deleted + \item Let $A_{ij} = (-1)^{i+j}\det(M_{ij})$, which is the $(i, j)$-cofactor + \item $\det(A) = a_{11}A_{11} + a_{12}A_{12} + ... + a_{1n}A_{1n}$ + + \item $\det(I) = 1$ + \item $A \xrightarrow{cR_i} B \implies \det(B) = c\det(A)$ + \subitem $I \xrightarrow{cR_i} E \implies \det(E) = c$ + \item $A \xrightarrow{R_1 \leftrightarrow R_2} B \implies \det(B) = -\det(A)$ + \subitem $I \xrightarrow{R_1 \leftrightarrow R_2} E \implies \det(E) = -1$ + \item $A \xrightarrow{R_i + cR_j} B \implies \det(B) = \det(A), i \neq j$ + \subitem $I \xrightarrow{R_i + cR_j} E \implies \det(E) = 1$ + \item $\det(EA) = \det(E)\det(A)$ + \end{itemize} + + Calculating determinants easier + \begin{itemize} + \item Let $A$ be square matrix. Apply Gaussian Elimination to get REF $R$ + \item $A \xrightarrow{E_1} \bullet \xrightarrow{E_2} \bullet ... \bullet \xrightarrow{E_k} R$ + \item $A \xleftarrow{E^{-1}_1} \bullet \xleftarrow{E^{-1}_2} \bullet ... \bullet \xleftarrow{E^{-1}_k} R$ + \item Since $E_i$ and $E^{-1}_k$ is type $II$ or $III$, $\det(E_i) = -1 / 1$ + \subitem $\det(A) = (-1)^t\det(R)$, where $t$ is no of type $II$ or $III$ operations + \item If $A$ is singluar, then $R$ has a zero row, and then $det(A) = 0$ + \item If A is invertible, then all rows of $R$ are nonzero + \subitem $\det(R) = a_{11}a_{22}...{a_nn}$, the product of diagonal entries. + \end{itemize} +\end{defn} + +\subsection{Recap} +\begin{itemize} + \item If A has a REF + \subitem If there is a zero row => Singular matrix + \subitem All rows are nonzero => invertible Matrix + \item If A is invertible, Using Gauss Jordan Elim $(A | I) \to (I | A^{-1})$ + \item +\end{itemize} + +\subsection{More about Determinants} + +\begin{defn}[Determinant Properties]\ \\ + $A$ is a Square Matrix +\begin{itemize} + \item $\det(A) = 0 \implies A$ is singular + \item $\det(A) \neq 0 \implies A$ is invertible + \item $\det(A) = \det(A^T)$ + \item $\det(cA) = c^n\det(A)$, where $n$ is the order of the matrix + \item If $A$ is triangular, $\det(A)$ product of diagonal entries + \item $\det(AB) = \det(A)\det(B)$ + \item $\det(A^{-1}) = [\det(A)]^{-1}$ +\end{itemize} + +Cofactor Expansion: +\begin{itemize} + \item To eavluate determinant using cofactor expansion, expand row/column with most no of zeros. +\end{itemize} +\end{defn} + +\subsection{Finding Determinants TLDR} +\begin{defn}[Finding Determinants]\ \\ + \begin{itemize} + \item If $A$ has zero row / column, $\det(A) = 0$ + \item If $A$ is triangular, $det(A) = a_{11}a_{22}...a_{nn}$ + \item If Order $n = 2 \to \det(A) = a_{11}a_{22} - a_{12}a_{21}$ + + \item If row/column has many 0, use cofactor expansion + \item Use Gaussian Elimination to get REF + \subitem $\det(A) = (-1)^t\det(R), t$ is no of type $II$ operations + \end{itemize} +\end{defn} + +\begin{defn}[Finding Inverse with Adjoint Matrix]\ \\ +\begin{itemize} + \item $\text{adj}(A) = (A_{ji})_{n\times n} = (A_{ij})^T_{n\times n}$ + \item $A^{-1} = [\det(A)]^{-1}\text{adj}(A)$ +\end{itemize} +\end{defn} + +\begin{defn}[Cramer's Rule] Suppose $A$ is an invertible matrix of order $n$ + \begin{itemize} + \item Liner system $Ax = b$ has unique solution + \item $x = \dfrac{1}{\det(A)}\begin{pmatrix}\det(A_1) \\ \vdots \\ det(A_n) \end{pmatrix}$, + \item $A_j$ is obtained by replacing the $j$th column in $A$ with $b$. + \end{itemize} +\end{defn} diff --git a/ma1522/ch_03.tex b/ma1522/ch_03.tex new file mode 100644 index 0000000..b98d1aa --- /dev/null +++ b/ma1522/ch_03.tex @@ -0,0 +1,12 @@ + +\subsection{Euclidian n-Spaces} + +\begin{defn}[Vector Definitions]\ \\ + \begin{itemize} + \item $n$-vector : $v = (v_1, v_2, ..., v_n)$ + \item $\vec{PQ} // \vec{P'Q'} \implies \vec{PQ} = \vec{P'Q'}$ + \item $|| \vec{PQ} || = \sqrt{(a_2 - a_1)^2 + (b_2 - b_1)^2}$ + \item $u + v = (u_1 + v_1, u_2 + v_2), u = (u_1, u_2), v = (v_1, v_2)$ + \item $n$-vector can be viewed as a row matrix / column matrix + \end{itemize} +\end{defn} diff --git a/ma1522/lec_01.tex b/ma1522/lec_01.tex deleted file mode 100644 index bbc8715..0000000 --- a/ma1522/lec_01.tex +++ /dev/null @@ -1,177 +0,0 @@ -\subsection{Linear Algebra} -\begin{itemize} - \item \textbf{Linear} The study of items/planes and objects which are flat - \item \textbf{Algebra} Objects are not as simple as numbers -\end{itemize} - -\subsection{Linear Systems \& Their Solutions} - -Points on a straight line are all the points $(x, y)$ on the $xy$ plane satisfying the linear eqn: $ax + by = c$, where $a, b > 0$ - -\subsubsection{Linear Equation} -Linear eqn in $n$ variables (unknowns) is an eqn in the form -$$ a_1x_1 + a_2x_2 + ... + a_nx_n = b$$ -where $a_1, a_2, ..., a_n, b$ are constants. - -\begin{note} -In a linear system, we don't assume that $a_1, a_2, ..., a_n$ are not all 0 -\begin{itemize} - \item If $a_1 = ... = a_n = 0$ but $b \neq 0$, it is \textbf{inconsistent} - - E.g. $0x_1 + 0x_2 = 1$ - - \item If $a_1 = ... = a_n = b = 0$, it is a \textbf{zero equation} - - E.g. $0x_1 + 0x_2 = 0$ - \item Linear equation which is not a zero equation is a \textbf{nonzero equation} - - E.g. $2x_1 - 3x_2 = 4$ - \item The following are not linear equations - \begin{itemize} - \item $xy = 2$ - \item $\sin\theta + \cos\phi = 0.2$ - \item $x_1^2 + x_2^2 + ... + x_n^2 = 1$ - \item $x = e^y$ - \end{itemize} -\end{itemize} -\end{note} - -In the $xyz$ space, linear equation $ax + by + cz = d$ where $a, b, c > 0$ represents a plane - -\subsubsection{Solutions to a Linear Equation} -Let $a_1x_1 + a_2x_2 + ... + a_nx_n = b$ be a linear eqn in n variables \\ -For real numbers $s_1+ s_2+ ... + s_n$, if $a_1s_1 + a_2s_2 + ... + a_ns_n = b$, then $x_1 = s_1, x_2 = s_2, x_n = s_n$ is a solution to the linear equation \\ -The set of all solutions is the \textbf{solution set}\\ -Expression that gives the entire solution set is the \textbf{general solution} - -\textbf{Zero Equation} is satified by any values of $x_1, x_2,... x_n$ - -General solution is given by $(x_1, x_2, ..., x_n) = (t_1, t_2, ..., t_n)$ - - -\subsubsection{Examples: Linear equation $4x-2y = 1$} -\begin{itemize} - \item x can take any arbitary value, say t - \item $x = t \Rightarrow y = 2t - \frac{1}{2}$ - \item General Solution: - $ - \begin{cases} - x = t & \text{t is a parameter}\\ - y = 2t - \frac{1}{2} - \end{cases} - $ - \item y can take any arbitary value, say s - \item $y = s \Rightarrow x = \frac{1}{2}s + \frac{1}{4}$ - \item General Solution: - $ - \begin{cases} - y = s & \text{s is a parameter}\\ - x = \frac{1}{2}s + \frac{1}{4} - \end{cases} - $ -\end{itemize} - -\subsubsection{Example: Linear equation $x_1 - 4x_2 + 7x_3 = 5$} - -\begin{itemize} - \item $x_2$ and $x_3$ can be chosen arbitarily, $s$ and $t$ - \item $x_1 = 5 + 4s -7t$ - \item General Solution: - $ - \begin{cases} - x_1 = 5 + 4s -7t \\ - x_2 = s & s, t \text{ are arbitrary parameters}\\ - x_3 = t \\ - \end{cases} - $ -\end{itemize} - - -\subsection{Linear System} -Linear System of m linear equations in n variables is -\begin{equation} -\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 \\ - a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_m \\ -\end{cases} -\end{equation} -where $a_{ij}, b$ are real constants and $a_{ij}$ is the coeff of $x_j$ in the $i$th equation - -\begin{note} Linear Systems - \begin{itemize} - \item If $a_{ij}$ and $b_i$ are zero, linear system is called a \textbf{zero system} - \item If $a_{ij}$ and $b_i$ is nonzero, linear system is called a \textbf{nonzero system} - \item If $x_1 = s_1, x_2 = s_2, ..., x_n = s_n$ is a solution to \textbf{every equation} in the system, then its a solution to the system - \item If every equation has a solution, there might not be a solution to the system - \item \textbf{Consistent} if it has at least 1 solution - \item \textbf{Inconsistent} if it has no solutions - \end{itemize} -\end{note} - - -\subsubsection{Example} -\begin{equation} -\begin{cases} -a_1x + b1_y = c_1 \\ -a_2x + b2_y = c_2 \\ -\end{cases} -\end{equation} - -where $a_1, b_1, a_2, b_2$ not all zero - -In $xy$ plane, each equation represents a straight line, $L_1, L_2$ - -\begin{itemize} - \item If $L_1, L_2$ are parallel, there is no solution - \item If $L_1, L_2$ are not parallel, there is 1 solution - \item If $L_1, L_2$ coinside(same line), there are infinitely many solution -\end{itemize} - -\begin{equation} -\begin{cases} -a_1x + b1_y + c_1z = d_1 \\ -a_2x + b2_y + c_2z = d_2 \\ -\end{cases} -\end{equation} - -where $a_1, b_1, c_1, a_2, b_2, c_2$ not all zero - -In $xyz$ space, each equation represents a plane, $P_1, P_2$ - -\begin{itemize} - \item If $P_1, P_2$ are parallel, there is no solution - \item If $P_1, P_2$ are not parallel, there is $\infty$ solutions (on the straight line intersection) - \item If $P_1, P_2$ coinside(same plane), there are infinitely many solutions - \item Same Plane $\Leftrightarrow a_1 : a_2 = b_1 : b_2 = c_1 : c_2 = d_1: d_2$ - \item Parallel Plane $\Leftrightarrow a_1 : a_2 = b_1 : b_2 = c_1 : c_2$ - \item Intersect Plane $\Leftrightarrow a_1 : a_2, b_1 : b_2, c_1 : c_2$ are not the same -\end{itemize} - -\subsection{Augmented Matrix} -$ - \begin{amatrix}{3} - a_{11} & a_{12} & a_{1n} & b_1 \\ - a_{21} & a_{12} & a_{2n} & b_2 \\ - a_{m1} & a_{m2} & a_{mn} & b_m \\ - \end{amatrix} -$ - -\subsection{Elementary Row Operations} -To solve a linear system we perform operations: -\begin{itemize} - \item Multiply equation by nonzero constant - \item Interchange 2 equations - \item add a constant multiple of an equation to another -\end{itemize} - -Likewise, for a augmented matrix, the operations are on the \textbf{rows} of the augmented matrix - -\begin{itemize} - \item Multiply row by nonzero constant - \item Interchange 2 rows - \item add a constant multiple of a row to another row -\end{itemize} - -To note: all these operations are revertible diff --git a/ma1522/lec_03.tex b/ma1522/lec_03.tex deleted file mode 100644 index 9bbb2c1..0000000 --- a/ma1522/lec_03.tex +++ /dev/null @@ -1,60 +0,0 @@ -\subsection{Review} - -\begin{align*} - I: & cR_i, c \neq 0 \\ - II: & R_i \Leftrightarrow R_j \\\ - III: & R_i \Rightarrow R_i + cR_j -\end{align*} - -Solving REF: -\begin{enumerate} - \item Set var -> non-pivot cols as params - \item Solve var -> pivot cols backwards - - \# of nonzero rows = \# pivot pts = \# of pivot cols -\end{enumerate} - -Gaussian Elimination -\begin{enumerate} - \item Given a matrix $A$, find left most non-zero \textbf{column}. If the leading number is NOT zero, use $II$ to swap rows. - \item Ensure the rest of the column is 0 (by subtracting the current row from tht other rows) - \item Cover the top row and continue for next rows -\end{enumerate} - -\subsection{Consistency} -\begin{defn}[Consistency]\ \\ - Suppose that $A$ is the Augmented Matrix of a linear system, and $R$ is a row-echelon form of $A$. - \begin{itemize} - \item When the system has no solution(inconsistent)? - \subitem There is a row in $R$ with the form $(0 0 ... 0 | \otimes)$ where $\otimes \neq 0$ - \subitem Or, the last column is a pivot column - \item When the system has exactly one solution? - \subitem Last column is non-pivot - \subitem All other columns are pivot columns - \item When the system has infinitely many solutions? - \subitem Last column is non-pivot - \subitem Some other columns are non-pivot columns. - \end{itemize} -\end{defn} - -\begin{note} Notations\ \\ - For elementary row operations - \begin{itemize} - \item Multiply $i$th row by (nonzero) const $k$: $kR_i$ - \item Interchange $i$th and $j$th rows: $R_i \leftrightarrow R_j$ - \item Add $K$ times $i$th row to $j$th row: $R_j + kR_i$ - \end{itemize} - \textbf{Note} - \begin{itemize} - \item $R_1 + R_2$ means "add 2nd row to the 1st row". - \item $R_2 + R_1$ means "add 1nd row to the 2st row". - \end{itemize} - - \textbf{Example} -$$ \begin{pmatrix} a \\ b \end{pmatrix} -\xrightarrow{R_1 + R_2} \begin{pmatrix} a + b \\ b \end{pmatrix} -\xrightarrow{R_2 + (-1)R_1} \begin{pmatrix} a + b \\ -a \end{pmatrix} -\xrightarrow{R_1 + R_2} \begin{pmatrix} b \\ -a \end{pmatrix} -\xrightarrow{(-1)R_2}\begin{pmatrix} b \\ a \end{pmatrix} - $$ -\end{note} diff --git a/ma1522/lec_04.tex b/ma1522/lec_04.tex deleted file mode 100644 index bedf46b..0000000 --- a/ma1522/lec_04.tex +++ /dev/null @@ -1,92 +0,0 @@ -\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. - \label{def:ltm} - \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 deleted file mode 100644 index 8cb9728..0000000 --- a/ma1522/lec_05.tex +++ /dev/null @@ -1,121 +0,0 @@ -\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 deleted file mode 100644 index 00e3c26..0000000 --- a/ma1522/lec_06.tex +++ /dev/null @@ -1,72 +0,0 @@ -\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} - -$A = \begin{pmatrix} - a_{11}&a_{12}&a_{13}\\ - a_{21}&a_{22}&a_{23}\\ - a_{31}&a_{32}&a_{33}\\ - a_{41}&a_{42}&a_{43} -\end{pmatrix}$, $E = \begin{pmatrix} - 1&0&0&0\\ - 0&1&0&0\\ - 0&0&c&0\\ - 0&0&0&1 -\end{pmatrix}(cR_3)$, $EA = \begin{pmatrix} - a_{11}&a_{12}&a_{13}\\ - a_{21}&a_{22}&a_{23}\\ - ca_{31}&ca_{32}&ca_{33}\\ - a_{41}&a_{42}&a_{43} -\end{pmatrix}$ diff --git a/ma1522/lec_07.tex b/ma1522/lec_07.tex deleted file mode 100644 index 695f449..0000000 --- a/ma1522/lec_07.tex +++ /dev/null @@ -1,107 +0,0 @@ -\begin{theorem} Main Theorem for Invertible Matrices \\ - Let $A$ be a square matrix. Then the following are equivalent - \begin{enumerate} - \item $A$ is an invertible matrix. - \item Linear System $Ax = b$ has a unique solution - \item Linear System $Ax = 0$ has only the trivial solution - \item RREF of $A$ is $I$ - \item A is the product of elementary matrices - \end{enumerate} -\end{theorem} - -\begin{theorem} Find Inverse - \begin{itemize} - \item Let $A$ be an invertible Matrix. - \item RREF of $(A | I)$ is $(I | A^{-1})$ - \end{itemize} - - How to identify if Square Matrix is invertible? - - \begin{itemize} - \item Square matrix is invertible - \subitem $\iff$ RREF is $I$ - \subitem $\iff$ All columns in its REF are pivot - \subitem $\iff$ All rows in REF are nonzero - \item Square matrix is singular - \subitem $\iff$ RREF is \textbf{NOT} $I$ - \subitem $\iff$ Some columns in its REF are non-pivot - \subitem $\iff$ Some rows in REF are zero. - \item $A$ and $B$ are square matrices such that $AB = I$ - \subitem then $A$ and $B$ are invertible - \end{itemize} -\end{theorem} - -\begin{defn}[LU Decomposition with Type 3 Operations]\ \\ - \begin{itemize} - \item Type 3 Operations: $(R_i + cR_j, i > j)$ - \item Let $A$ be a $m \times n$ matrix. Consider Gaussian Elimination $A \dashrightarrow R$ - \item Let $R \dashrightarrow A$ be the operations in reverse - \item Apply the same operations to $I_m \dashrightarrow L$. Then $A = LR$ - \item $L$ is a \hyperref[def:ltm]{lower triangular matrix} with 1 along diagonal - \item If $A$ is square matrix, $R = U$ - \end{itemize} - - Application: - \begin{itemize} - \item $A$ has LU decomposition $A = LU$, $Ax = b$ i.e., $LUx = b$ - \item Let $y = Ux$, then it is reduced to $Ly = b$ - \item $Ly = b$ can be solved with forward substitution. - \item $Ux = y$ is the REF of A. - \item $Ux = y$ can be solved using backward substitution. - - \end{itemize} -\end{defn} - -\begin{defn}[LU Decomposition with Type II Operations]\ \\ - \begin{itemize} - \item Type 2 Operations: $(R_i \leftrightarrow R_j)$, where 2 rows are swapped - \item $A \xrightarrow[]{E_1} \bullet \xrightarrow[]{E_2}\bullet \xrightarrow[E_3]{R_i \iff R_j}\bullet \xrightarrow[]{E_4}\bullet \xrightarrow[]{E_5} R$ - \item $A = E^{-1}_1E^{-1}_2E^{}_3E^{-1}_4E^{-1}_5R$ - \item $E_3A = (E_3E^{-1}_1E^{-1}_2E_3)E^{-1}_4E^{-1}_5R$ - \item $P = E_3, L = (E_3E^{-1}_1E^{-1}_2E_3)E^{-1}_4E^{-1}_5, R = U$, $PA = LU$ - - \end{itemize} -\end{defn} - -\begin{defn}[Column Operations]\ \\ - \begin{itemize} - \item Pre-multiplication of Elementary matrix $\iff$ Elementary row operation - \subitem $A \to B \iff B = E_1E_2...E_kA$ - \item Post-Multiplication of Elementary matrix $\iff$ Elementary Column Operation - \subitem $A \to B \iff B = AE_1E_2...E_k$ - \item If $E$ is obtained from $I_n$ by single elementary column operation, then - \subitem $I \xrightarrow[]{kC_i}E \iff I \xrightarrow[]{kR_i}E$ - \subitem $I \xrightarrow[]{C_i \leftrightarrow C_j}E \iff I \xrightarrow[]{R_i \leftrightarrow R_j}E$ - \subitem $I \xrightarrow[]{C_i + kC_j}E \iff I \xrightarrow[]{R_j + kR_i}E$ - \end{itemize} -\end{defn} - -\subsection{Determinants} - -\begin{defn}[Determinants of $2 \times 2$ Matrix]\ \\ - \begin{itemize} - \item Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ - \item $\det(A) = |A| = ad - bc$ - \item $\det(I_2) = 1$ - \item $A \xrightarrow{cR_i} B \implies \det(B) = c\det(A)$ - \item $A \xrightarrow{R_1 \leftrightarrow R_2} B \implies \det(B) = -\det(A)$ - \item $A \xrightarrow{R_i + cR_j} B \implies \det(B) = \det(A), i \neq j$ - \end{itemize} - Solving Linear equations with determinants for $2 \times 2$ - \begin{itemize} - \item $x_1 = - \dfrac{\begin{vmatrix} b_1 & a_{12} \\ b_2 & a_{22} \end{vmatrix}} - {\begin{vmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}}$, $x_2 = - \dfrac{\begin{vmatrix} a_{11} & b_1 \\ a_{21} & b_2 \end{vmatrix}} - {\begin{vmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}}$ - \end{itemize} -\end{defn} - -\begin{defn}[Determinants of $3 \times 3$ Matrix]\ \\ - \begin{itemize} - \item Suppose $A$ is invertible, then there exists EROs such that - \item $A \xrightarrow{ero_1} A_1 \rightarrow ... \rightarrow A_{k-1} \xrightarrow{ero_k}A_k = I$ - \item Then $\det(A)$ can be evaluated backwards. - \subitem E.g. $A \xrightarrow{R_1 \leftrightarrow R_3} \bullet \xrightarrow{3R_2} \bullet \xrightarrow{R_2 + 2R_4} I \implies det(A) = 1 \to 1 \to \frac{1}{3} \to -\frac{1}{3}$ - \end{itemize} -\end{defn} diff --git a/ma1522/lec_08.tex b/ma1522/lec_08.tex deleted file mode 100644 index e69de29..0000000 diff --git a/ma1522/lec_09.tex b/ma1522/lec_09.tex deleted file mode 100644 index e69de29..0000000 diff --git a/ma1522/lec_10.tex b/ma1522/lec_10.tex deleted file mode 100644 index e69de29..0000000 diff --git a/ma1522/lec_11.tex b/ma1522/lec_11.tex deleted file mode 100644 index e69de29..0000000 diff --git a/ma1522/lec_12.tex b/ma1522/lec_12.tex deleted file mode 100644 index e69de29..0000000 diff --git a/ma1522/lec_13.tex b/ma1522/lec_13.tex deleted file mode 100644 index e69de29..0000000 diff --git a/ma1522/preamble.tex b/ma1522/preamble.tex index 1b3ed83..b9c2723 100644 --- a/ma1522/preamble.tex +++ b/ma1522/preamble.tex @@ -22,6 +22,7 @@ \let\implies\Rightarrow \let\impliedby\Leftarrow \let\iff\Leftrightarrow +\let\vec\overrightarrow \newcommand{\HRule}[1]{\rule{\linewidth}{#1}}