線性代數-陳良華老師(選)
===
:::success
用書 : linear algebra and its applications 6th
:::

:::spoiler 目錄
[TOC]
:::
# 9/7
:::spoiler 筆記圖檔by BluePumpkin


:::

## 矩陣Matrix

| [2X3]Metrix| Column 1 | Column 2 | Column 3 |
| --- | -------- | -------- | -------- |
| **Row1** | A~11~ | A~12~ | A~13~ |
| **Row2** | A~21~ | A~22~ | A~23~ |

$<A>_{ij}$=the entry in the ==i==Row and ==j==Column at Metrix==A==

## Operation in Matrices
1. ***Sum difference*** ==(same size)==

ex:
A=

| 2 | 3 | 4 |
| -------- | -------- | -------- |
| 1 | 2 | 1 |

B=

| 0 | 2 | 7 |
| -------- | -------- | -------- |
| 1 | -3 | 5 |

A+B=

| 2+0 | 3+2 | 4+7 |
| -------- | -------- | -------- |
| 1+1 | 2-3 | 1+5 |

- A-B同理

2. ***scalar mutiple***

ex:
E=

| -2 | 4 | 3+2i |
| -------- | -------- | -------- |
| 10 | 6 | -7 |

4E=

| -8 | 16 | 12+8i |
| -------- | -------- | -------- |
| 40 | 24 | -28 |

- 除法同理

3. ***Mutiplication***

<ruby>A．B=C<rp>(</rp><rt>mx~~r．r~~xn=mxn</rt><rp>)</rp></ruby>
r相消--->**equal and cancell**

$$ <C>~ij~=\sum_{k=1}^r(<A>_{ik}．<B>_{kj})
$$

ex:
A=

| ==2== | ==3== |
| -------- | -------- |
| 1 | 2 |

B=

| ==0== | 2 | 7 |
| -------- | -------- | -------- |
| ==1== | -3 | 5 |

A．B=2x~~2~~．~~2~~x3=2x3Matrix

| ==2x0+3x1== | 2x2+3x-3 | 2x7+3x5 |
| -------- | -------- | -------- |
| 1x0+2x1 | 1x2+2x-3 | 1x7+2x5 |

4. ***properties***

- A+B = B+A
- A+(B+C) = (A+B)+C
- A(BC) = (AB)C $\not=$ ==BA==C
- AB != BA (證明看圖)
- if A=B then AC=BC,CA=CB
- if AB=AC B$\not=$C(證明看圖)
- AB=O $\not=$> A=O B=O(證明看圖)

5. ***corollary***
- (A+b)^2^ $\not=$ A^2^+2AB+B^2^<br>因為 (A+b)^2^ = A^2^+AB+BA+B^2^ 且 AB $\not=$ BA<br>所以 (A+B)^2^ $\not=$ A^2^+2AB+B^2^
- (A+B)(A-B) 同理 , AB $\not=$ BA 故不可相消

## Transpose(T)
$<A^T>~ij~ = <A>~ji~$

ex
A=
| ==2== | ==1== | ==5== |
| --- | --- | --- |
| 3 | 4 | 6 |

A^T^=
| ==2== | 3 |
| --- | --- |
| ==1== | 4 |
| ==5== | 6 |

- (A^T^)^T^ = A
- (A$\pm$B)^T^ = A^T^ $\pm$ B^T^
- (KA)^T^ = KA^T^
- (AB)^T^ = A^T^B^T^ , (ABC)^T^ = C^T^B^T^A^T^

## Square Matrix
Column = Row = n
means a Square Matrix ==order n==

- Zero Matrix($O$)

| 0 | 0 |
|:--- | --- |
| 0 | 0 |

- Diagonal Matrix
- diag(2,-1)

| ==2== | 0 |
|:-----:|:------:|
| 0 | ==-1== |
- diag(4,0,2,3)

| ==4== | 0 | 0 | 0 |
|:-----:|:-----:|:-----:|:-----:|
| 0 | ==0== | 0 | 0 |
| 0 | 0 | ==2== | 0 |
| 0 | 0 | 0 | ==3== |

- unit matrix($I$)
- $I^T = I$
- $AI = A = IA$
- ($I$)=diag(1,1,1,...)

| ==1== | 0 | 0 |
|:-----:|:-----:|:-----:|
| 0 | ==1== | 0 |
| 0 | 0 | ==1== |

- Triangular matrix
- upper triangular($u$)

| 5 | 4 | 6 |
|:-----:|:-----:|:---:|
| ==0== | 1 | 3 |
| ==0== | ==0== | 7 |

- lower triangular ($L$)

| 1 | ==0== | ==0== |
|:---:|:-----:|:-----:|
| 5 | 7 | ==0== |
| 6 | 5 | 8 |

- $u^T = L$
- $u*u = u$
- $L^T = u$
- $L*L = L$

# 9/14

>發現新大陸，直接去514巷影印店買講義跟考古題[name=BluePumpkin]

###### tags: `1111` `2022`
<style>
.navbar-brand::after { content: " × FJUMIIA"; }
</style>