MATH 3210 Class Notes 02-26-2025

notevector-spaces

Theorem 1

If $V_i$ is finite-dimensional, then $V_1\times \cdots \times V_n$ is finite dimensional and,

$$\dim (V_1\times \cdots \times V_n)=\dim V_1+\cdots +\dim V_n.$$

Remark 1

There is a canonical map $\Gamma :V_1\times \cdots \times V_n\to V_1+\cdots +V_n$, given by $\Gamma (x_1,\dots ,x_n)=x_1+\cdots +x_n$.

Theorem 2

$V_1+\cdots +V_n$ is a direct sum if and only if $\Gamma$ is injective.

Example 3

Suppose $T:V\to W$ is linear.

The graph of $T$ is the subspace of $V\times W$ given by graph of $T=\left\{(v,Tv):v\in V\right\}.$

Quotient Spaces

Notation

Let $v\in V$ and $U\subseteq V$, then $v+U$ is the subset of $V$ defined by $v+U\left\{v+u:u\in U\right\}.$

Example 1

Let $U=\left\{(x,2x):x\in \mathbb{R}\right\}\subseteq {\mathbb{R}}^2$. What is $(5,1)+U$?

$v+U$ is a translate of $U$.

Defined Quotient Space

Proposition

Suppose $U$ is a subspace of $V$ and $v,w\in V$, then $v-w\in U⟺v+U=w+U⟺(v+U)\cap (w+U)\ne \varnothing .$

First Isomorphism Theorem of Vector Spaces

Theorem

Suppose $U$ is a subspace of $V$ such that ${}^V{/}_U$ is finite-dimensional, then

$$V\simeq U\times ({}^V{/}_U).$$

Duality

Dual Vector Space

For some vector space $V$ over the field $\mathbb{F}$, we have that the dual vector space $V^{\prime }=\mathcal{L}(V,\mathbb{F})$.

Example

If $V=P(\mathbb{R})$, $\mathbb{F}=\mathbb{R}$, then $\phi (p)=5p^{\prime \prime }(0)\in (P(\mathbb{R}){)}^{\prime }$.