Linear System

1. System: a system has input and output (function, transformation, operator)
2. Linear system $\Leftrightarrow$ two conditions:
   • Persevering Multiplication: $f(c\bm{v})=cf(\bm{v})$
   • Persevering Addition: $f(\bm{u}+\bm{v})=f(\bm{u})+f(\bm{v})$
   • In this class: when the input and output are vectors
   • Including: transpose, derivative, integral (from $a$ to $b$)

Vector

1. A vector $\bm{v}$ is a set of numbers
2. Component: the entries of a vector:
   • the $i\text{-}th$ component of vector $\bm{v}$ refers to $v_i$
3. Vector Set: a vector set can contain infinite elements
   • $\mathcal{R}^{n}$ the set of all vectors with $n$ components
4. Scalar Multiplication:
5. Vector Addition:
6. Properties of Vector / Definition of Vectors $\Leftrightarrow$
   • $\bm{u}+\bm{v}=\bm{v}+\bm{u}$
   • $(\bm{u}+\bm{v})+\bm{w}=\bm{v}+(\bm{u}+\bm{w})$
   • There is an element $\bm{0}$ in $\mathcal{R}^{n}$ such that $\bm{0}+\bm{u}=\bm{u}$
   • There is an element $\bm{0}$ in $\mathcal{R}^{n}$ such that $\bm{u}'+\bm{u}=\bm{0}$
   • $1\bm{u}=\bm{u}$
   • $(ab)\bm{u}=a(b\bm{u})$
   • $a(\bm{u}+\bm{v})=a\bm{v}+a\bm{u}$
   • $(a+b)\bm{u}=a\bm{u}+b\bm{u}$

Linear system and system of linear equations

1. Linear System $\Leftrightarrow$ System of Linear Equations
   • From a linear system $\rightarrow$ a system of linear equations
2. Standard (unit) vector: only one element is "1", the rest elements are "0"
   • standard vectors are the building blocks of any vector

Matrix

1. Denotation

   • If the matrix has m rows and n columns , we say the size of the matrix is $m$ by $n$, written $m\times n$
   • We use $\mathcal M_{m\times n}$ to denote the set that contains all matrices whose size is $m\times n$
   • first row, then column

2. Index of component: the scalar in the $i\text{-}th$ row and $j\text{-}th$ column is called $(i,j)\text{-}entry$ of the matrix

   $$ \it A=\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots\ & a_{mn} \\ \end{bmatrix} \\

   \text{or}\\\it{A} =[\bm{a}_1,\bm{a}_2,...,\bm{a}_n] $$

3. Properties

   • $\it A+\it B=\it B+\it A$
   • $(\it A+\it B)+\it C=\it A+(\it B+\it C)$
   • $(st)\it A=s(t\it A)$
   • $s(\it A+\it B)=s\it A+s\it B$
   • $(s+t)\it A=s\it A+t\it B$

4. Well-known Matrices

   • Square Matrix: $m=n$

     $\ \ \ \ \ \ \ \ \ \ \ \ \ \textcolor{red}{n}\\\textcolor{blue}{m}\begin{bmatrix}\bold 2,&\bold 0,&\bold 0\\\bold0,& 1,&0\\\bold0,&0,& 2\end{bmatrix} \text{where\ }\textcolor{blue}{m}=\textcolor{red}{n}$

     • Only square matrix has diagonal

       • Upper Triangular Matrix

         $\begin{bmatrix}2,&3,&5\\\bold 0,&1,&-1\\\bold 0,&\bold 0,&1\end{bmatrix}$

       • Lower Triangular Matrix

         $\begin{bmatrix}2,&\bold 0,&\bold 0\\3,&1,&\bold 0\\-2,&1,&1\end{bmatrix}$

   • Diagonal Matrix: all non-diagonal elements are "0"

     $\begin{bmatrix}\bold 2,&0,&0\\0,&\bold 1,&0\\0,&0,&\bold 2\end{bmatrix}$

   • Identity Matrix: denoted by I (any size) or $\bm I_n$

     $\it I_3 =\begin{bmatrix}1,&0,&0\\0,&1,&0\\0,&0,& 1\end{bmatrix}$

   • Zero Matrix: denoted by $\it O$ (any size) or $\it{O}_{m\times n}$

     $\it{0}_{2\times 3} =\begin{bmatrix}0,&0,&0\\0,&0,&0\end{bmatrix}$

   • Transpose: If $\it{A}$ is an $m\times n$ matrix, $\it{A^T}$is an $n\times m$ matrix whose $(i,j)\text{-}entry$ is the $(j,i)$of A

     • $\it{(A^T)^T}=\it A$
     • $\it{(sA)^T=sA^T}$
     • $\it{(A+B)^T=A^T+B^T}$
     • This is a linear system 🙂

   • Symmetric Matrix $\Leftrightarrow\it{A^T=A}$ a symmetric matrix must be a square matrix because of the size