MATH 323 Lecture 9

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Euclidean Vector Space

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

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

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

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

Scalar Multiplication

Multiply all components by scaling factor $\alpha$

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

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

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

Vector Addition and Subtraction

Apply operation to corresponding components

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

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

Matrix Space

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

Let $V$ be a set with the operation of addition and scalar multiplication:

Closure Properties

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

Axioms

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

Examples

Sets

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

Continuous Functions

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

Polynomials

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

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

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

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

Theorem

$V$ is vector space, $x\in V$ , then

$0\cdot x=0$
$x+y=0\implies y=-x$ (additive inverse is unique)
$(-1)x=-x$

Proof

$x=1\cdot x=(1+0)x=x+0x\rightarrow x-x=0x=0$
$-x=-x+0=-x+(x+y)=0+y=y$
$0=0x=(1+(-1))x=x+(-1)x\rightarrow (-1)x=x$

Subspace

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

Example

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