# Linear Algebra
## Introduction
1) Geometry
-linear
-to study geometry w. linearity
-In
**2D: lines
3D: planes
nD: hyperplanes**(out of the box 超過想像範圍)

2) Abstract algebra
-Algebra is a study of basic mathematical structures
e.g. **groups, rings, fields**, etc.
-Linear alg. studies one of the structures called the **vector space**
-followed by logical deduction from basic definitions,we can derive some important theorems

3) Applied mathematics
-Linear alg is widely used in applied science
-Live mechanics: differential equations (ODE,PDE)
-Linear programming was developed during WW2.
-Recently applied in **image processing** and **computer graphics**.

Application:
-Linear programming
-[Least square solution](https://textbooks.math.gatech.edu/ila/least-squares.html#:~:text=So%20a%20least%2Dsquares%20solution,difference%20b%20%E2%88%92%20Ax%20is%20minimized.)
-Differential equation
-Image Processing
-Filter
-Deep learning ~ [LeNet 5](https://en.wikipedia.org/wiki/LeNet)

## Chapter 1
### 1.1 Intro
The central problem of linear alg is solution of linear equations. The most important and the simplest case is when the
**# of unknowns = # of equations**.
There are **2** ways to solve linear equations.
A) **Elimination method** ([Gaussian elimination](https://en.wikipedia.org/wiki/Gaussian_elimination))
B) **Determinants** ([Cramer's Rule](https://www.purplemath.com/modules/cramers.htm))

4 aspects that we should look into:
I ) The geometry of linear equations
n = 2, n = 3 ->higher dimension (hyperplane,LA has any answers/solutions)
II ) The interpretation of elimination is a factorization of the coefficient matrix.
III) Irregular/singular cases <no unique solution>
-> no solution / infinitely many solutions
IV) # of operations to solve the system by elimination

### 1.2 Geometry of Linear Equations
2x - y = 1
x + y = 5
<Approach 1> Row picture -> 2 lines in the plane

~> x=2 y=3 : intersection of 2 lines

<Approach 2> Column picture
$$
x\begin{bmatrix} 2\\
1\\
\end{bmatrix}+y\begin{bmatrix} -1\\
1\\
\end{bmatrix}=\begin{bmatrix} 1\\
5\\
\end{bmatrix}
$$
2x1 (2d space)
->to find the linear combination of
$$
\begin{bmatrix} 2\\1\\\end{bmatrix} \begin{bmatrix}-1\\1\\\end{bmatrix}
$$
to form a column vector
\begin{bmatrix} 1\\5\\\end{bmatrix}

parallelogram x=2, y=3

Note: a vector is a n*1 array w. n real numbers, c~n~'s,
\begin{bmatrix} c1\\c2\\.\\.\\cn\\\end{bmatrix}
in the text, we usually write it as (c~1~,c~2~,...,c~n~)
$$
α\begin{bmatrix} c1\\c2\\.\\.\\cn\\\end{bmatrix} = \begin{bmatrix}αc1\\αc2\\.\\.\\αcn\\\end{bmatrix} , α ∈ ℝ
, \begin{bmatrix} c1\\c2\\.\\.\\cn\\\end{bmatrix}+\begin{bmatrix} d1\\d2\\.\\.\\dn\\\end{bmatrix} = \begin{bmatrix}c1+d1\\c2+d2\\.\\.\\cn+dn\\\end{bmatrix}
$$
same dimension

y ∈ ℝ, y ∈ ℝ^2^
$$
y=\begin{bmatrix} y1\\y2\\\end{bmatrix}
$$