Lecture 1: Course Introduction

1.1 Outline Introduction

Introduction

Basic Content:
Determinant
Matrix
Theme:
sets of vectors
System of equations
Application:
Eigenvalues;
At least five points;
quadratic form;
With eigenvalues, quadratic forms can be analyzed;
To study shapes in space, quadratic techniques are needed -> maximum and minimum values in graphs -> minimum value problems;

Concept: Research directions and tools

Research Directions:
The research content isvector: Vector
The number of vectors equals their dimensions;
Determinant:
$$\begin{vmatrix}\mathrm{m}&&\mathrm{n}\\\mathrm{a}&&\mathrm{b}\end{vmatrix}=\mathrm{mb}-\mathrm{an}$$
Research Tools:
1. Linear Operations -> Multiplication and Addition;
2. Point product operations -> In linear algebra, point product operations are essentially still linear operations;
$(a_{1}a_{2})(\begin{matrix}b_{1}\\b_{2}\end{matrix})=a_1b_1+a_2b_2$
$(a_1a_2)\begin{pmatrix} b_1 & c_1 \\ b_2 & c_2 \end{pmatrix}=(a_{1}b_{1}+a_{2}b_{2},a_{1}c_{1}+a_{2}c_{2})$
$\begin{pmatrix} a_1 & a_2 \\ b_1 & b_2 \end{pmatrix}(c_1c_2)=\begin{pmatrix} a_1c_1+a_2c_2 \\ b_1c_1+b_2c_2 \end{pmatrix}$

Concept: Research methods

Concept:
Core -> linear transformation;
Matrix -> Represents system information
Data in the matrix cannot be manipulated arbitrarily, otherwise system information will be corrupted;
Linear transformation:
Analysis:
$(\begin{matrix}1&0\\0&-1\end{matrix})(\begin{matrix}1\\1\end{matrix})=(\begin{matrix}1\\-1\end{matrix})$
Where $(\begin{matrix}1&0\\0&-1\end{matrix})$ is a matrix, corresponding to the function f in advanced mathematics
Where $(\begin{matrix}1\\1\end{matrix})$ represents the variable, corresponding to x in advanced mathematics;
Where $(\begin{matrix}1\\-1\end{matrix})$ corresponds to the junction, corresponding to the y in advanced mathematics;
The input is a vector;
Example:
Example: Symmetric transformation
Matrix: $(\begin{matrix}1&0\\0&-1\end{matrix})$
Example: Scaling transformation
Matrix: Scale to the right $(\begin{matrix}2&0\\0&1\end{matrix})$
Matrix: Scaling upward $(\begin{matrix}1&0\\0&2\end{matrix})$
Example: Shear transformation
Matrix: $(\begin{matrix}1&-1\\0&1\end{matrix})$
Matrix: $(\begin{matrix}-1&1\\0&1\end{matrix})$

Analysis: Systems of equations and linear transformations

Example: $\begin{cases}x_{1}+2x_{2}=3\\4x_{1}+7x_{2}=10\end{cases}$
Traditional method: Elimination method
Linear transformation:
Because $x_{1}+2x_{2}$ and $4x_{1}+7x_{2}=10$ -> are the result of dot products;
So it can be transformed into a linear transformation: $\left.\left(\begin{matrix}1&2\\4&7\end{matrix}\right.\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}3\\10\end{matrix}\right)$
Therefore, based on this linear transformation, find the $x_1x_2$ of it
Solution: Multiply both sides of the equivalence by the inverse of the matrix. Assume $A=(\begin{matrix}1&2\\4&7\end{matrix})$, then multiply both sides by $A^{-1}$
So we get $Ax=B$ -> $AA^{-1}x=BA^{-1}$ -> $x=BA^{-1}$
That is, the solution for x can be obtained;

Concept: What is dot product?

Linear transformation angle:
Formula:
$$\overbrace{\begin{bmatrix}1&-2\end{bmatrix}}^{\text{Transform}}\underbrace{\begin{bmatrix}4\\3\end{bmatrix}}_{\begin{array}{c}\text{Vector}\\\text{}\end{array}}=4\cdot1+3\cdot-2$$
Illustration: