MATH 323 Lecture 9

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Euclidean Vector Space

$\overrightarrow{x}=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\in {\mathbb{ℝ}}^2$

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

$\Vert \overrightarrow{x}\Vert =\sqrt{{x_1}^2+{x_2}^2}$ is Euclidean length of a 2D vector.

Arbitrary: $\Vert \overrightarrow{x}\Vert =\sqrt{{x_1}^2+{x_2}^2+\dots +{x_n}^2}$

Scalar Multiplication

Multiply all components by scaling factor $\alpha$

$\alpha \overrightarrow{x}=\left[\begin{array}{c}\alpha x_1\\ \alpha x_2\end{array}\right]$ scales vector

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

Arbitrary: $\alpha \overrightarrow{x}=\left[\begin{array}{c}\alpha x_1\\ \alpha x_2\\ ⋮\\ \alpha x_n\end{array}\right]$

Vector Addition and Subtraction

Apply operation to corresponding components

$\begin{array}{rl}\overrightarrow{x}+\overrightarrow{y}& =\left[\begin{array}{c}{x}_1+y_1\\ x_2+y_2\end{array}\right]\\ \overrightarrow{x}-\overrightarrow{y}& =\left[\begin{array}{c}{x}_1-y_1\\ x_2-y_2\end{array}\right]\end{array}$

Arbitrary: $\overrightarrow{x}\pm \overrightarrow{y}=\left[\begin{array}{c}{x}_1\pm y_1\\ x_2\pm y_2\\ ⋮\\ x_n\pm y_n\end{array}\right]$

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

1. $x\in V⟹\alpha x\in V\mathrm{\forall }\alpha \in \mathbb{ℝ}$
2. $x,y\in V⟹x+y\in V$

Axioms

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

Examples

Sets

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

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 $n\in \mathbb{\mathbb{N}}$.

A polynomial $P(x)=a_0+a_{1}x+a_2{x}^2+\dots +a_n{x}^n$

degree of $P$ (denoted $\mathrm{deg}P(x)$) 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

1. $0\cdot x=0$
2. $x+y=0⟹y=-x$ (additive inverse is unique)
3. $(-1)x=-x$

Proof

1. $x=1\cdot x=(1+0)x=x+0x\to x-x=0x=0$
2. $-x=-x+0=-x+(x+y)=0+y=y$
3. $0=0x=(1+(-1))x=x+(-1)x\to (-1)x=x$

Subspace

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

Example

$S=\left\{\left[\begin{array}{c}{x}_1\\ 3x_1\end{array}\right]\right|x_1\in \mathbb{ℝ}\}$ is a subspace of ${\mathbb{ℝ}}^2$ since it satisfies the closure properties