# 線性代數

## Matrices and Gaussian Elimination

### 1.1 Introduction

The central problem of linear algebra is the solution of linear equations.
The most important and the simplest case is when the number of unknowns equals to the number of equations.
There are two ways to solve linear equations.
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>(1)The method of elimination. (高斯消去法)
>(2)Determinants. (行列式)
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Four aspects that we should look into:
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>(1) The geometry of linear equations.(n=2,n=3 → higher dimension)
>(2) The interpretation of elimination is a factorization of the coefficient matrix.
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>* Ax = b
>* A = LU
>* A^T/A^-1
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>(3) Irregular of Singular cases ***no unique solutions*** → no solution or inginitely many solutions
>(4) Numbers of options to solve the system by elimination.
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### 1.2 Geometry of Linear Equations

$$\left\{\begin{array}{l}2x - y = 1\\ x + y = 5\\\end{array}\right.$$

< approach *I*> row picture &rarr; two lines in the plane
< approach *II*> column picture

$$x\left[\begin{array}{cc}2\\1\\\end{array}\right]+y\left[\begin{array}{cc}-1\\1\\\end{array}\right] = \left[\begin{array}{cc}1\\5\\\end{array}\right]$$

$\Rightarrow$ To find the linear combination of $\left[\begin{array}{cc}2\\1\\\end{array}\right]$,$\left[\begin{array}{cc}-1\\1\\\end{array}\right]$ to form a column vector $\left[\begin{array}{cc}1\\5\\\end{array}\right]$
< parallelogram > x = 2 , y = 3
**Note:** A vector is an n\*1 arrray with n real numbers.
$$Cn's:\left[\begin{array}{c}C_{1}\\C_{2}\\C_{3}\\\vdots\\C_{n}\\\end{array}\right]$$
**Example**:

$$\left\{\begin{array} 2u + v + w = 5\\ 4u - 6v = -2\\-2u + 7v + 2w = 9\\\end{array}\right.$$

#### <*I*> row picture

&rarr;three planes in the space($\mathbb{R}^{3}$)
**Note:** In n dimensions, a line requires n-1 equations.
**Question:** How to extend into n dimensions?

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>* Each equation represents a plane (with higher dimension)
>* The first equation produces an (n-1)-dimensional plane in n dimensions
>* The second equaion produces another one
>* They instersect in a smaller set of (n-2)-dimension
>n &rarr; n-1 &rarr; n-2 &rarr; ... &rarr; 3(space) &rarr; 2(plane) &rarr; 1(line) &rarr; point
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#### <*II*> column picture

$u\left[\begin{array}{cc}2\\4\\2\\\end{array}\right] + v\left[\begin{array}{cc}1\\-6\\7\\\end{array}\right] + w\left[\begin{array}{cc}1\\0\\2\\\end{array}\right] = \left[\begin{array}{cc}5\\-2\\9\\\end{array}\right]$

RHS is a linear combination of 3 colomn vectors.
$\Rightarrow$ Solution to equations :
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>intersections of n planes (row picture)
>coefficients in combination of vectors (column picture)
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***THE SINGULAR CASE***
<*I*> row picture : In 3D case, three planes do not intersect.

- case 1 : two planes are parallel$$\left\{\begin{array}{l}2u + v + w = 5\\4u + 2v + 2w = 9\\\end{array}\right .$$
- case 2 : all three planes are prependicular ($\bot$) to the board $$\left\{\begin{array}{l} u + v + w = 2\cdots (1)\\2u + 3w = 5\cdots (2)\\3u + v + 4w = 6\cdots (3)\\\end{array}\right .$$
LHS : (1)+(2)=(3)
RHS : 2 + 5 $\neq$ 6
$\Rightarrow$ inconsistent!
- case 3 : three planes have a whole line in common
- case 4 : three planes are parallel