MATH 414 Lecture 1

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Vector Spaces

A vector space $V$ is a nonempty set of objects (called vectors ) that is closed over the following operations:

• addition: $\mathrm{\forall }\overrightarrow{u},\overrightarrow{v}\in V\overrightarrow{u}+\overrightarrow{v}\in V$
• scalar multiplication: $\mathrm{\forall }\overrightarrow{v}\in V\mathrm{\forall }c\in \mathbb{ℂ}c\overrightarrow{v}\in V$

These operations have the following properties:

• additive commutativity
• additive associativity
• additive identity $\overrightarrow{0}$
• additive inverse $-\overrightarrow{u}$
• scalar multiplicative identity $1$
• scalar multiplicative associativity
• left- and right-distributivity

Examples

• Euclidean Space: ${\mathbb{ℝ}}^n$
• Complex Euclidean Space: ${\mathbb{ℂ}}^n$
• Matrices: $M_{m\times n}\left(\mathbb{ℝ}\right)$ and $M_{m\times n}\left(\mathbb{ℂ}\right)$

Sequence Spaces

Elements of the form: $\overrightarrow{x}=⟨x_1,x_2,\dots ,x_n,\dots ⟩$, where $x_i\in \mathbb{ℝ}$ or $x_i\in \mathbb{ℂ}$.

Bilateral: $\overrightarrow{x}=⟨\dots ,x_{-2},x_{-1},x_0,x_1,x_2,\dots ⟩$, where $x_i\in \mathbb{ℝ}$ or $x_i\in \mathbb{ℂ}$.

Sequence spaces can be considered digital versions of a signal:

• Label an axis with units
• Signal sampling takes values $x_i$ at each point $i$.

Function Spaces

Let $C[a,b]=\{f:[a,b]\to \mathbb{ℂ}\mid f\text{is continuous on}[a,b]\}$

Think of $f$ as a rule that assigns a scalar to each $x\in [a,b]$.

Polynomial Spaces

${{𝒫}}_n=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{1,x,\dots ,x^n\}$, so $\mathrm{d}\mathrm{i}\mathrm{m}\left({{𝒫}}_n\right)=n+1$

Inner Product Space

Let $V$ be a vector space over complex scalars $\mathbb{ℂ}$. The inner product of two vectors $\overrightarrow{u},\overrightarrow{v}\in V$, denoted $⟨\overrightarrow{u},\overrightarrow{v}⟩$ is a scalar. For complex numbers, we define the inner product as

$⟨\overrightarrow{u},\overrightarrow{v}⟩=u_1{\overline{v}}_1+u_2{\overline{v}}_2$

Where $\overline{z}=a-bi$ if $z=a+bi$ represents the conjugate

Properties:

• Conjugate symmetry: $\overline{⟨\overrightarrow{u},\overrightarrow{v}⟩}=⟨\overrightarrow{v},\overrightarrow{u}⟩$
• Homogeneneous