yadunut/nus
1
0
Fork 0
Code Issues Pull Requests Actions Projects Releases Wiki Activity

feat: 1522 rename to chapters

Browse Source
This commit is contained in:
Yadunand Prem 2024-02-28 15:53:52 +08:00
parent a8a385633f
commit 9162dcb311
No known key found for this signature in database
18 changed files with 726 additions and 675 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
ma1522/1522 Notes.pdf
View File

Binary file not shown.

53
ma1522/1522 Notes.tex
View File

@ -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}

260
ma1522/lec_02.tex → ma1522/ch_01.tex
View File

@ -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}

446
ma1522/ch_02.tex Normal file
View File

@ -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}

12
ma1522/ch_03.tex Normal file
View File

@ -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}

177
ma1522/lec_01.tex
View File

@ -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

60
ma1522/lec_03.tex
View File

@ -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}

92
ma1522/lec_04.tex
View File

@ -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}

121
ma1522/lec_05.tex
View File

@ -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}

72
ma1522/lec_06.tex
View File

@ -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}
$A = \begin{pmatrix}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
a_{31}&a_{32}&a_{33}\\
a_{41}&a_{42}&a_{43}
\end{pmatrix}$, $E = \begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&c&0\\
0&0&0&1
\end{pmatrix}(cR_3)$, $EA = \begin{pmatrix}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
ca_{31}&ca_{32}&ca_{33}\\
a_{41}&a_{42}&a_{43}
\end{pmatrix}$

107
ma1522/lec_07.tex
View File

@ -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}

0
ma1522/lec_08.tex
View File

0
ma1522/lec_09.tex
View File

0
ma1522/lec_10.tex
View File

0
ma1522/lec_11.tex
View File

0
ma1522/lec_12.tex
View File

0
ma1522/lec_13.tex
View File

1
ma1522/preamble.tex
View File

@ -22,6 +22,7 @@
\let\implies\Rightarrow
\let\impliedby\Leftarrow
\let\iff\Leftrightarrow
\let\vec\overrightarrow
\newcommand{\HRule}[1]{\rule{\linewidth}{#1}}