###### tags: `ntust` `Math` `note` `ntust`
# Linear Algebra

:::info
值得一覽：
1. [講義](https://web.ntnu.edu.tw/~40247038S/Linear%20Algebra/)
2. [影片](https://www.youtube.com/watch?v=csgNflj69-Y&list=PLybg94GvOJ9En46TNCXL2n6SiqRc_iMB8&index=1&ab_channel=ProfessorDaveExplains)
:::

# Matrices ans Systems of Equations

## 1.1 System of Linear Equations

### [**linear equation(線性方程式)**](https://zh.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84)
- A linear in n unknows is an equation of the form
$$ a_1x_1+a_2x_2+...+a_nx_n = b $$
- where $a_1,a_2,...a_n$ and $b$ are real numbers and $x_1,x_2,...x_n$ are variables. A linear system of m equations in unknows is then a system of the form.

$$a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1$$

$$a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_2$$

$$. $$

$$. $$

$$. $$

$$a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_m$$

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### **Linear Systems(線性系統)**
1. <font color = red>**Unique Solution**(有唯一解的恰定方程組)</font>
- Intersection Lines(兩條線交於一點)

2. <font color = red>**No solution**(解不存在的超定方程組)</font>
* Parallel Lines(兩條線平行)

3. <font color = red>**Infinite Solution** or **coincide** (有無窮多解的欠定方程組（也被通俗地稱為不定方程組))</font>
* Same Line(兩條線重合)
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### **Equivalent Systems(等價系統)**
- <font color = #0000FF>Two systems of equations involving the same variables are said to be **eqivalent** if they have the same set.</font>
>如果兩個 linear system 具有相同數目的未知數，並且有相同的解(必須是一樣的集合)。那我們說這兩個 system Equivalent(相等)。

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### **Strict Triangular Form(嚴格三角方陣)**
- <font color = #0000FF>A system is said to be in stict triangle form if, in the kth eqution, the coeffcients of the first $k-1$ variables are all zero and the coefficient of $x_K$ is nozereo (k=1,...,n).</font>
>如果在一個 system 中，第 k 個等式的前 k - 1 個變數的係數都是 0，而第 k 個變數的係數不為 0。那就稱這個形式為 strict triangular form。


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### **Augmented matrix(增廣矩陣)**
- 將原本的 + , x, = 等符號省略, 只留係數在方陣中
- 必須以相同的順序放置常數
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### **Elementary Row Operations(基本列運算)**
以下三種計算方式不會影響到 system 本身的解。所有藉由這個方式衍生的 system 都跟原來的 system equivalent。

1. 將兩列(等式)互相交換
2. 將某一列(等式)乘上一個常數
3. 將某列(等式)替換為自己與另一列(等式)的總合
- 這三種方法通常是用來取得 **strict triangular form** 而使用的


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## 1.2 [Row Echelon Form(階梯形矩陣)](https://zh.wikipedia.org/wiki/%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5)
A matrix is said to be row echelon form if
1. The first nozero entry in each nonzero row is 1.
2. If row $k$ does not consist entirely of zeros, the number of leading zero entires in row $k+1$ is greater than the number of leading zero enrties in row $k$.
3. If there are rows whose entries are all zero, they are below the rows having nonzero entries.

>- 線性代數中，一個矩陣如果符合下列條件的話，我們稱之為**列階梯形矩陣**或列**梯形式矩陣**（英語：Row Echelon Form）：
> 1. 所有非零列（矩陣的列至少有一個非零元素）在所有全零列的上面。即全零列都在矩陣的底部。
> 2. 非零列的首項係數（leading coefficient），也稱作軸元(pivot或pivot element)，即最左邊的首個非零元素，嚴格地比上面列的首項係數更靠右（某些版本會要求非零列的首項係數必須是1）。
> 3. 首項係數所在行，在該首項係數下面的元素都是零（前兩條的推論）。
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### [Overdetermined Systems](超定系統)
- **Overdetermined** 就是指等式的數目比未知數還要多的 system，通常這種情況下會是無解(但也有可能有解)。

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### [Underdetermined Systems](欠定方程)
- **Underdetermined** 就是指等式的數目比未知數還要少的 system，通常這種情況下會有無限多組解(也有可能無解)，但是不可能只有一組解。
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### [Reduced Row Echelon Form(簡化列階梯形矩陣)](https://zh.wikipedia.org/wiki/%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5)
- A matrix is in row echelon form if
1. The matrix is in row echelon form.
2. The forst nonzero entry in each row is **the only nonzero entry in its column**.

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### Homogeneous Systems(齊次系統)

- **Homogeneous** systems are always **consistent**
- Homogeneous systems **at least one solution**.
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- [video](https://www.youtube.com/watch?v=JlJWyWJARRU&ab_channel=patrickJMT)
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## 1.3 [Matrix Arithmetic(矩陣運算)](https://online.stat.psu.edu/statprogram/reviews/matrix-algebra/arithmetic)

::: spoiler `Matrix Notation`

$$
\left[\begin{matrix}
a_{11}&a_{12}&_{...}&a_{1n}\\
a_{21}&a_{22}&_{...}&a_{2n}\\
_{...}\\
a_{m1}&a_{m2}&_{...}&a_{mn}
\end{matrix}\right]
$$
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- [video](https://www.youtube.com/watch?v=y6bVhgmy2rw&ab_channel=ProfessorDaveExplains)
:::

::: spoiler `Vectors`
- [連結](https://zh.wikipedia.org/wiki/%E8%A1%8C%E5%90%91%E9%87%8F%E8%88%87%E5%88%97%E5%90%91%E9%87%8F)

:::

:::spoiler `Equality`
- Tow $m*n$ matrices $A$ and $B$ are said to be equal if $a_{ij} = b_{ij}$ for each $i$ and $j$.
:::

:::spoiler `Sclar Multiplication`
- [連結](https://zh.wikipedia.org/wiki/%E6%A0%87%E9%87%8F%E4%B9%98%E6%B3%95)
- If $A$ is an $m*n$ matrix and $\alpha$ is a ascalar, then $\alpha a$ is the $m*n$ matrix whose$(i,j)$ entry is $\alpha a_{ij}$.
:::

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### Matrix addition
-