---
# System prepended metadata

title: Subspaces
tags: [Linear Algebra]

---

---
tags: Linear Algebra
---

## Subspaces
It's **my lecture notes**, [**1.3 Subspaces**](https://www.youtube.com/watch?v=UnUCe3cIE34&list=PLj6E8qlqmkFtjxknKFtdxc1_SxNBXgpbo&index=3), [Linear Algebra 1, NYCU OCW](https://ocw.nycu.edu.tw/?course_page=all-course%2Fcollege-of-science%2Fam%2F%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8%E4%B8%80-linear-algebra-i-%E6%87%89%E7%94%A8%E6%95%B8%E5%AD%B8%E7%B3%BB-%E8%8E%8A%E9%87%8D%E8%80%81%E5%B8%AB)

## Facebook 嘅討論
- [2023-12-05 13:49](https://www.facebook.com/groups/978651839991898/posts/1056975408826207/)
- [2023-12-02 00:19](https://www.facebook.com/groups/978651839991898/posts/1055163385674076/)

## Review The Definition of Vector Spaces
A vector space $V$ over a field $F$ consists of a set on which two operations,
**addition** and **scalar multiplication**, are satisfied with the following axioms

(VS-1) $x + y \in V$ whenever $x,\ y \in V$

(VS 0) $\alpha x \in V$ whenever $\alpha \in F\ and\ x \in V$

(VS 1) $x + y = y + x\;\; \forall\ x,\ y \in V$ (commutative)

(VS 2) $(x + y) + z = x + (y + z)\;\; \forall\ x,\ y,\ z \in V$ (associative)

(VS 3) $\exists\ \mathbf{0} \in V$ such that $x + \mathbf{0} = x,\;\: x \in V$

(VS 4) For each $x \in V, \; \exists\:y\in V$ such that $x + y = \mathbf{0}$

(VS 5) For each $x \in V$, $\mathbf{1} \cdot x=x\;, \mathbf{1} \in F$

(VS 6) $(ab)x = a(bx)\;\; \forall\; a,\ b \in F\ and\ x \in V$ (distributive)

(VS 7) $a(x + y) = ax + ay\;\; \forall\; a,\ b \in F\ and\ x \in V$ (distributive)

(VS 8) $(a + b)x = ax + bx\;\; \forall\; a,\ b \in F\ and\ x \in V$ (distributive)

## The Definition of Subspaces
Let $V$ be a Vector Space over a field $F$, then $W \subset V$ is called a subspace of $V$ if $W$ is a Vector Space over the $F$ under the operations of $+$ and $\cdot$ defined on $V$.

## Theorem 1.3
$W \subset V$, where $V$ is a Vector Space, Then $W$ is a subspace of $V$ and the following statements are true. 
1. $\mathbf{0} \in W$
2. $x + y \in W$ whenever $x,\ y \in W$
3. $cx \in W$ whenever $c \in F,\ x \in W$

## Theorem 1.4
Any intersection of subspace of a vector space $V$ is a subspace of $V$.

## Exercises

The following problems are taken from the textbook, [Linear Algebra, 4th Edition by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence](https://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/dp/0130084514)

#### Section 1.3 Subspaces
10. For $W_1$
    
    Verify Theorem 1.3 Part 1
    
    $a_1 + a_2 + \cdots + a_n = 0$ 
    $\implies$ $\mathbf{0} \in F^n$
    
    just reminder: $(1, -1, \cdots, 1, -1)$ also $\in F^n$
    
    Verify Theorem 1.3 Part 2
    
    Take a vector $\vec{c} = (c_1, \cdots, c_n) \in W_1$, a vector $\vec{d} = (d_1, \cdots, d_n) \in W_1$
  
    $\vec{c} + \vec{d} = (c_1, \cdots, c_n) + (d_1, \cdots, d_n) = (c_1 + d_1,\  \cdots,\ c_n + d_n)$
    
    since $c_1 + \cdots + c_n = 0$, $d_1 + \cdots + d_n = 0$
    $c_1 + d_1 +  \cdots + c_n + d_n = (c_1 + \cdots + c_n) + (d_1 + \cdots + d_n) = 0$
  
    Verify Theorem 1.3 Part 3
    
    Take a vector $\vec{c} = (c_1, \cdots, c_n) \in W_1$
    
    $k \cdot \vec{c} = (kc_1, \cdots, kc_n)$
    
    $kc_1 + \cdots + kc_n$ = $k(c_1 + \cdots + c_n) = 0$
    
    hence, $W_1$ is a subspace of $F^n$
    
    ~~Other Proof~~
    $a_1 + a_2 + \cdots + a_n = 0$ means $\vec{a} \cdot \vec{1} = 0$, so span$\{\vec{a}, \vec{1}\}$ is a plane in $F^n$, it's a subspace
    
    For $W_2$
    
    Verify Theorem 1.3 Part 1
    
    It collects all vectors in $F^n$ which the sum of all components of the vector equals $1$, but the sum of the zero vector must be zero, so $\mathbf{0} \notin F^n$

    Verify Theorem 1.3 Part 2

    Take a vector $\vec{c} = (c_1, \cdots, c_n) \in W_2$, a vector $\vec{d} = (d_1, \cdots, d_n) \in W_2$
  
    $c_1 + \cdots + c_n = 1$, $d_1 + \cdots + d_n = 1$
  
    $\vec{c} + \vec{d} = (c_1, \cdots, c_n) + (d_1, \cdots, d_n) = (c_1 + d_1,\  \cdots,\ c_n + d_n)$
    
    $c_1 + d_1 +  \cdots + c_n + d_n = (c_1 + \cdots + c_n) + (d_1 + \cdots + d_n) = 2 \neq 1$
    
    Verify Theorem 1.3 Part 3
    
    Take a vector $\vec{c} = (c_1, \cdots, c_n) \in W_2$
    
    $k \cdot \vec{c} = (kc_1, \cdots, kc_n)$
    
    $kc_1 + \cdots + kc_n$ = $k(c_1 + \cdots + c_n) = k \neq 1$
    
    hence, $W_2$ isn't a subspace of $F^n$
    