# MATH 323 Lecture 9

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## Euclidean Vector Space

${\displaystyle {\vec {x}}={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}\in \mathbb {R} ^{2}}$

In general, euclidean space is defined by any ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle \|{\vec {x}}\|={\sqrt {{x_{1}}^{2}+{x_{2}}^{2}}}}$ is Euclidean length of a 2D vector.

Arbitrary: ${\displaystyle \|{\vec {x}}\|={\sqrt {{x_{1}}^{2}+{x_{2}}^{2}+\dots +{x_{n}}^{2}}}}$

### Scalar Multiplication

Multiply all components by scaling factor ${\displaystyle \alpha }$

${\displaystyle \alpha \,{\vec {x}}={\begin{bmatrix}\alpha \,x_{1}\\\alpha \,x_{2}\end{bmatrix}}}$ scales vector

• stays parallel
• ${\displaystyle \alpha >0}$ → result vector is same direction as original
• ${\displaystyle \alpha <0}$ → result vector is in opposite direction.

Arbitrary: ${\displaystyle \alpha \,{\vec {x}}={\begin{bmatrix}\alpha \,x_{1}\\\alpha \,x_{2}\\\vdots \\\alpha \,x_{n}\end{bmatrix}}}$

Apply operation to corresponding components

{\displaystyle {\begin{aligned}{\vec {x}}+{\vec {y}}&={\begin{bmatrix}x_{1}+y_{1}\\x_{2}+y_{2}\end{bmatrix}}\\{\vec {x}}-{\vec {y}}&={\begin{bmatrix}x_{1}-y_{1}\\x_{2}-y_{2}\end{bmatrix}}\end{aligned}}}

Arbitrary: ${\displaystyle {\vec {x}}\pm {\vec {y}}={\begin{bmatrix}x_{1}\pm y_{1}\\x_{2}\pm y_{2}\\\vdots \\x_{n}\pm y_{n}\end{bmatrix}}}$

## Matrix Space

${\displaystyle \mathbb {R} ^{m\times n}}$ represents space of ${\displaystyle m\times n}$ matrices

Let ${\displaystyle V}$ be a set with the operation of addition and scalar multiplication:

### Closure Properties

1. ${\displaystyle x\in V\implies \alpha \,x\in V\quad \forall \alpha \in \mathbb {R} }$
2. ${\displaystyle x,y\in V\implies x+y\in V}$

### Axioms

1. ${\displaystyle x+y=y+x\quad \forall x,y\in V}$ (commutativity of addition)
2. ${\displaystyle (x+y)+z=x+(y+z)\quad \forall x,y,z\in V}$ (associativity of addition)
3. ${\displaystyle \exists 0\in V~:~x+0=x\quad \forall x\in V}$ (additive identity)
4. ${\displaystyle \forall x\in V\exists y\in V~:~x+y=0}$ (additive inverse; ${\displaystyle y}$ denoted ${\displaystyle -x}$)
5. ${\displaystyle \alpha (x+y)=\alpha \,x+\alpha \,y\quad \forall \alpha \in \mathbb {R} \quad x,y\in V}$ (distributive 1)
6. ${\displaystyle (\alpha +\beta )x=\alpha \,x+\beta \,x\quad \forall \alpha ,\beta \in \mathbb {R} }$ (distributive 2)
7. ${\displaystyle (\alpha \,\beta )x=\alpha (\beta \,x)\quad \forall \alpha ,\beta \in \mathbb {R} }$ (associativity of multiplication)
8. ${\displaystyle \exists 1\in V~:~1\cdot x=x\quad \forall x\in V}$ (multiplicative identity)

### Examples

#### Sets

${\displaystyle W=\left\{(a,2)~|~a\in \mathbb {R} \right\}\subset \mathbb {R} ^{2}}$ represents the line ${\displaystyle y=2}$. ${\displaystyle W}$ is not space since it does not satisfy the closure properties: let ${\displaystyle \alpha =100}$; ${\displaystyle (100\,a,200)\not \in W}$

#### Continuous Functions

${\displaystyle f,g\in C[a,b]}$ represents the space of continuous functions on the interval ${\displaystyle [a,b]}$. All axioms and closure properties are satsified for this space.

#### Polynomials

Focus on ${\displaystyle x^{n}}$ for ${\displaystyle n\in \mathbb {N} }$.

A polynomial ${\displaystyle P(x)=a_{0}+a_{1}\,x+a_{2}\,x^{2}+\dots +a_{n}\,x^{n}}$

degree of ${\displaystyle P}$ (denoted ${\displaystyle \deg P(x)}$) is ${\displaystyle n}$ (max power of ${\displaystyle x}$)

Let ${\displaystyle P_{n}}$ represent all polynomials of degree < ${\displaystyle n}$. ${\displaystyle P_{n}}$ satisfies all properties and axioms and is therefore spaaaaaace!

### Theorem

${\displaystyle V}$ is vector space, ${\displaystyle x\in V}$, then

1. ${\displaystyle 0\cdot x=0}$
2. ${\displaystyle x+y=0\implies y=-x}$ (additive inverse is unique)
3. ${\displaystyle (-1)x=-x}$

#### Proof

1. ${\displaystyle x=1\cdot x=(1+0)x=x+0x\rightarrow x-x=0x=0}$
2. ${\displaystyle -x=-x+0=-x+(x+y)=0+y=y}$
3. ${\displaystyle 0=0x=(1+(-1))x=x+(-1)x\rightarrow (-1)x=x}$

### Subspace

For a space ${\displaystyle V}$, ${\displaystyle W\subseteq V}$ is a subspace of ${\displaystyle V}$ if it also satisfies the closure properties

#### Example

${\displaystyle S=\left\{{\begin{bmatrix}x_{1}\\3\,x_{1}\end{bmatrix}}~{\big |}~x_{1}\in \mathbb {R} \right\}}$ is a subspace of ${\displaystyle \mathbb {R} ^{2}}$ since it satisfies the closure properties