---
title: Ch1-1
tags: Linear algebra
GA: G-77TT93X4N1
---

# Chapter 1 extra note 1
> Vector space

## Youtube

* [What is a Vector Space? (Abstract Algebra)](https://youtu.be/ozwodzD5bJM?si=PcvFWVno-auUXDUj)

## Selected lecture notes:

### Vector space

A vector space $V$ over a field $F$ is a collection of vectors together with vector addition and scalar multiplication that satisfy the following $8$ properties: For all ${\bf u, v, w}\in V$ and $\alpha, \beta\in F$,
1. Commutativity: ${\bf u + v} = {\bf v + u}$;
2. Associativity: ${\bf( u + v) + w} = {\bf u + (v +w)}$;
3. Zero vector: There exists ${\bf 0}\in V$ such that ${\bf u + 0} = {\bf u}$;
4. Additive inverse: For every vector ${\bf u}\in V$, there exists a vector ${\bf u'}\in V$ s.t. ${\bf u + u'} = {\bf 0}$;
5. Multiplicative identity: $1{\bf v} = {\bf v}$;
6. Multiplicative associativity: $(\alpha\beta){\bf v} = \alpha(\beta{\bf v})$;
7. $\alpha ({\bf u+v}) = \alpha{\bf u} + \alpha{\bf v}$;
8. $(\alpha + \beta) {\bf u} = \alpha{\bf u} + \beta{\bf u}$.

#### Theorems

9. If ${\bf u+v} = {\bf u+w}$, then ${\bf v} = {\bf w}$.
10. ${\bf 0}$ is unique.
11. $0 \cdot {\bf u} ={\bf 0}$.
12. $\alpha \cdot {\bf 0} ={\bf 0}$.
13. Additive inverse is unique, and equals to $(-1)\cdot{\bf u}$.

#### Proof of the theorems:

* Proof of 9:
> pf:
> Since ${\bf u}\in V$, (by 4) there exists ${\bf u'}\in V$ such that ${\bf u + u'} = {\bf 0}$.
> Then we add ${\bf u'}$ to both side of ${\bf u+v} = {\bf u+w}$.
> The left hand side gives
> $$
> {\bf (u+v) + u'} ={\bf (v+u) + u'}={\bf v + (u+ u')}={\bf v + 0} = {\bf v},
> $$
> where we have used 1, 2 and 3.
> Similarly, the right hand side gives ${\bf (u+w) + u'} ={\bf w}$.
> Making both sides equal gives ${\bf v} = {\bf w}$.

* Proof of 10:
> pf:
> Suppose there exists ${\bf 0_1}\in V$ and ${\bf 0_2}\in V$ such that ${\bf u+0_1}={\bf u+0_2} = {\bf u}$ for all ${\bf u}\in V$.
> Since ${\bf 0_1}$ and ${\bf 0_2}\in V$, we must have ${\bf 0_2+0_1}= {\bf 0_2}$ and ${\bf 0_1+0_2} ={\bf 0_1}$. Therefore, using 1),
> $$
> {\bf 0_1} = {\bf 0_1+0_2} = {\bf 0_2+0_1} = {\bf 0_2}.
> $$
> So the zero vector is unique.

* Proof of 11:
> pf:
> Given ${\bf u}\in V$, using 3 and 8, we have
> $$
> 1\cdot {\bf u} + {\bf 0}= 1\cdot {\bf u} = (1+0)\cdot {\bf u} = 1\cdot {\bf u} + 0\cdot {\bf u}.
> $$
> Then by 9), we have $0\cdot {\bf u}= {\bf 0}$.

* Proof of 12:
> pf:
> Given $\alpha\in F$, let ${\bf u}\in V$, using 7 and 3,
> $$
> \alpha\cdot{\bf u}+\alpha\cdot{\bf 0} = \alpha({\bf u+ 0}) = \alpha\cdot{\bf u}= \alpha\cdot{\bf u} + {\bf 0}.
> $$
> Then by 9), we have $\alpha\cdot{\bf 0}= {\bf 0}$.

* Proof of 13-1:
> pf:
> Let ${\bf u}\in V$, suppose there exists ${\bf u'_1}\in V$ and ${\bf u'_2}\in V$ such that ${\bf u + u'_1}={\bf 0}$ and ${\bf u + u'_2}={\bf 0}$. Since ${\bf 0}$ is unique by 10), we have
> $$
> {\bf u + u'_1} = {\bf u + u'_2}.
> $$
> Then by 9), we have ${\bf u'_1} = {\bf u'_2}$.

* Proof of 13-2:
> pf:
> Let ${\bf u}\in V$, using 5, 8 and 11,
> $$
> {\bf u} + (-1)\cdot {\bf u} = 1\cdot {\bf u} + (-1)\cdot {\bf u} = (1+ (-1)){\bf u} = 0\cdot{\bf u} = {\bf 0}.
> $$
> So its additive inverse is $(-1)\cdot {\bf u}$.

### Examples

#### Examples of vector space

* With usual vector addition and scalar multiplication
* $\mathbb{R}^n$: n-dimensional Euclidean space.
* $\mathbb{P}_n$: Polynomials of degree at most $n$.
* $M_{m\times n}$: $m\times n$ matrices.
* $C([0, 1])$: Continuous function defined on $[0, 1]$.
* The set of all the even functions on $\mathbb{R}$. ($f(x) = f(-x)$, $x\in\mathbb{R}$.)
* a) $V = \{x | x\in\mathbb{R}^+\}$ with $\oplus$ and $\odot$ defined as
$$
x \oplus y = xy, \quad \alpha \odot y = y^{\alpha}.
$$
* b) $V = \{(x_1, x_2, x_3) | x_1, x_2, x_3\in\mathbb{R}^+, x_1+x_2+x_3=1\}$, $F=\mathbb{R}$, with $\oplus$ and $\odot$ defined as
$$
\begin{align}
(x_1, x_2, x_3) \oplus (y_1, y_2, y_3) &= \frac{1}{x_1y_1+x_2y_2+x_3y_3}(x_1y_1, x_2y_2, x_3y_3),\\
\alpha \odot (x_1, x_2, x_3) &= \frac{1}{x_1^{\alpha}+x_2^{\alpha}+x_3^{\alpha}}(x_1^{\alpha}, x_2^{\alpha}, x_3^{\alpha}).
\end{align}
$$

#### Examples of non-vector space

* The set of all polynomials of degree **exactly** $n$.
* The set of all continuous and positive functions on $[0, 1]$.
* c) Let $V=\mathbb{R}^2$ with $\oplus$ and $\odot$ defined as
$$
(a_1, a_2)\oplus (b_1, b_2) = (a_1+b_1, a_2b_2), \quad \alpha\odot (a_1, b_1) = (\alpha a_1, b_1).
$$
* d) Let $V=\mathbb{R}^2$ with usual $\odot$ while $\oplus$ is defined as
$$
(a_1, a_2)\oplus (b_1, b_2) = (a_1+2b_1, a_2+2b_2).
$$
* e) Let $V=\mathbb{R}^2$ with usual vector $\oplus$ while $\odot$ is defined as
$$
\alpha\odot (a_1, b_1) = \begin{cases} (\frac{a_1}{\alpha}, \frac{a_2}{\alpha}), & \mbox{if $\alpha\ne 0$,}\\
(0, 0), & \mbox{if $\alpha=0$.}
\end{cases}
$$