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tags: Linear Algebra - HungYiLee
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Linear Algebra - Lec 18~20: Subspace and Basis
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[TOC]

# [課程網站](http://speech.ee.ntu.edu.tw/~tlkagk/courses_LA18.html)

# [Lecture 18: Subspace](https://www.youtube.com/watch?v=pXtXnY2b2-E&list=PLJV_el3uVTsNmr39gwbyV-0KjULUsN7fW&index=18)


## Subspace v.s. Span

- the **span of a vector set is a subspace**
- 
- $u = c_1w_1+...+c_kw_k, v = c_1'w_1+...+c_k'w_k$
- **反過來**其實也成立，**每一個 Subspace 都是由某個 vector set 產生出來的 Span**，下節課會講

## Null Space
- The null space of a matrix $A$ is the **solution set** of $Ax=0$, denoted as $Null(A)$
- $Null(A)$ is a **subspace**.

## Column Space and Row Space

### Column Space
- column space of matrix $A$ is the **span of its columns**, it is denoted as $\text{Col}\, A$
- $A\in\mathbb R^{m\times n}\implies \text{Col}\,A=\{Av:v\in\mathbb R^n\}$
- if matrix $A$ represents a function, then $\text{Col}\,A$ is the **range of the function**
- 因為 $\text{Col}\,A$ 此時代表 $Ax$ 的**所有可能的輸出**所形成的集合

### Row Space
- row space of matrix $A$ is the **span of its rows**, denoted as $\text{Row}\,A$
- $\text{Row}\,A=\text{Col}\,A^T$

### Column Space = Range

## RREF v.s. Column Space & Row Space

Original Matrix $A$ v.s. its RREF $R$ (Review of Lec 10)
- Columns:
- The relations between the columns are the same (review)
- **But** the span of the columns are different
- $\text{Col}\,A\neq\text{Col}\,R$
- Rows:
- The relations between the rows are changed
- **But** the span of the rows are the same
- $\text{Col}\,A=\text{Col}\,R$

## Review: Consistent

## Conclusion
Subspace is Closed under **Addition and Multiplication**.

# [Lecture 19: Basis](https://www.youtube.com/watch?v=GB48DyvC14o&list=PLJV_el3uVTsNmr39gwbyV-0KjULUsN7fW&index=19)

## Outline
- what is a basis **for a subspace**?
- confirming that a set is a basis for a subspace

## Basis
- Let $V$ be a **nonzero subspace** of $\mathbb R^n$. A <font color="red">basis $B$</font> for $V$ is a <font color="blue">linearly independent</font> <font color="green">generation set of $V$</font>
- $B$ is generation set of $V$ 表示 <font color="green">$V=Span\,B$</font>

### Example

## Basis of Column Space
- The **pivot column** of matrix $A$ form a basis for its **column space**
- pivot column 即矩陣 $A$ 的 RREF 的 leading entry 所在 columns (Lec 8)

## Property of Span

## Basis 最重要的 3 個定理 (Theorem) & Dimension
- The number of vectors in a basis for a nonzero subspace $V$ is called dimension of $V$, denoted $dim(V)$

## 先證明第三個定理

- Any two bases of a subspace $V$ contain **the same number** of vectors

## n 維空間的 basis 就有 n 個 vector，dimension 就是 n

### Example for finding dim V

- 這裡可複習一下 Lec 8 的 Parametric Representation
- 把 **linear system 的解** 給算出來，就可以得到它的 **basis**
- ***Q: 是指 column space 的 basis 嗎??? 還是 Null Space 的 basis ??? 還是都可以啊???***
- 看到這樣的式子，換句話說就是 Span of 這三個 vector 會包含 $b$

## 證明第一個定理

懶得看ㄌ，略過

要先證明 reduction theorem

### Reduction Theorem

- 所有的 generation set 裡面都包含了 basis (basis 的 vector 數目一定小於等於 generation set 的 vector 數目)

證明: 將 generation set 裡的所有 vector 都排成一個 matrix $A$，而這個 matrix 的 column space 的 basis 就是它的 pivot column，得證。

- 記憶法：**所有的 generation set 心中都有一個 basis**

## 證明第二個定理
- 若 basis 有 $k$ 個 vector，則在 subspace 中找不到多於 $k$ 個 independent 的 vector

### Extension Theorem
- Given an independent vector set $S$ in the subspace, $S$ can be extended to a basis by adding more vectors.

- 記憶法：**indepdent vector set: 我不是 basis，就是正在成為 basis 的路上**

證明:

## Summary


## Confirming that a set is a basis

有點難，所以我們需要另外一種方式來檢查是否為 basis

## Another way to confirm if is basis

- 此時只需要檢查其中一個條件，basis 就成立

證明:

Example:

# [Lecture 20: Column Space, Null Space, Row Space](https://www.youtube.com/watch?v=aW0JVmpIxas&list=PLJV_el3uVTsNmr39gwbyV-0KjULUsN7fW&index=20)

## Review of Column/Null/Row Space (Lec 18)

$A\in\mathbb R^{m\times n}$

- Column Space
- $Col\,A\in\mathbb R^m$
- 是 $A$ 的 columns 的 linear combination 的集合
- 是 $A$ 這個 linear system 的 co-domain 的 range
- Null Space
- $Null\,A\in\mathbb R^n$
- 是 $Ax=0$ 的所有解 $x$ 形成的集合
- 必定包含 zero vector
- Row Space
- $Row\,A=Col\,A^T$

- 那如何找到這些 space 的 **basis**、計算它們的 **dimension** 呢？

## Column Space: Basis and Dimension
- basis: **pivot columns** of $A$ form a basis of $Col\,A$
- **Dim(Col A)** = number of pivot columns of $A$ = $rank\,A$，所以 $A$ 的 rank 又等於 $A$ 的 column space 的 dimension

### Review of Rank:
- Maximum number of independent columns
- Number of pivot columns
- Number of non-zero rows in $RREF(A)$
- Number of basic variables

Now we also know that $rank\,A$ is
- **Dim (Col A)**: dimension of column space of $A$
- Dimension of the range of $A$

## Null Space: Basis and Dimension
- Basis
1. 把 $A$ 做 RREF
2. 轉換成 parametric representation (review Lec 8)
3. 再轉換成向量 linear combination 的形式，那些**和 free variable 相乘的 vectors** 就是 basis
- Dimension
- $Dim(Null\,A)\\ = \text{number of free variables}\\=Nullity\,A\\=n-rank(A)$

如此我們知道
- $A$ 的 column space 的 dimension 是 $rank(A)$
- $A$ 的 null space 的 dimension 是 $nullity(A)$

由此可知，$Dim(Col\,A)+Dim(Null\,A)=n$，即
- **$A$ 的 column space 和 null space 的 dimension 加起來會是 $n$**

## Row Space: Basis and Dimension
- **Basis: non-zero rows of RREF(A)**
- **Dimension: Rank A (居然和 column space 的 dimension 一樣！)**

## Rank A (revisit)

## 矩陣做 transpose 後，rank 仍一樣
- $Rank\,A=Rank\,A^T$

## Dimension Theorem
- **Dim of Range + Dim of Null = Dim of Domain**
- domain 是該 linear system 所有可能的 input 形成的集合；是所有可能的 output 形成的集合???

## Summary