MATH 414 Lecture 1

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Vector Spaces

A vector space $V$ is a nonempty set of objects (called vectors ^[1]) that is closed over the following operations:

addition: $\forall {\vec {u}},{\vec {v}}\in V\quad {\vec {u}}+{\vec {v}}\in V$
scalar multiplication: $\forall {\vec {v}}\in V\quad \forall c\in \mathbb {C} \quad c\,{\vec {v}}\in V$

These operations have the following properties:

additive commutativity
additive associativity
additive identity ${\vec {0}}$
additive inverse $-{\vec {u}}$
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: ${\vec {x}}=\left\langle x_{1},x_{2},\ldots ,x_{n},\ldots \right\rangle$ , where $x_{i}\in \mathbb {R}$ or $x_{i}\in \mathbb {C}$ .

Bilateral: ${\vec {x}}=\left\langle \ldots ,x_{-2},x_{-1},x_{0},x_{1},x_{2},\ldots \right\rangle$ , 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 $C\left[a,b\right]=\left\{f:\left[a,b\right]\to \mathbb {C} \mid f\ {\mbox{is continuous on}}\ \left[a,b\right]\right\}$

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

Polynomial Spaces

${\mathcal {P}}_{n}=\mathrm {Span} \left\{1,x,\ldots ,x^{n}\right\}$ , so $\mathrm {dim} \left({\mathcal {P}}_{n}\right)=n+1$

Inner Product Space

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