02 - Vector Spaces

Vector Space $V$

Vector Space $V$ is defined over a field $\mathbf{F}$ where

$$(V,+,0)\text{ is an abelian group}$$

combined with scalar multiplication

$$F\times V\to V$$

such that for all $a,b\in \mathbf{F}$ and $v,w\in V$

$$\begin{array}{rlrl}& (1)& & a(v+w)=av+aw\\ & (2)& & (a+b)v=av+bv\\ & (3)& & (ab)v=a(bv)\\ & (4)& & 1\cdot v=v\end{array}$$

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Property of Direct Sum

Let $V$ be a vector space
Suppose $V=W_1\oplus \cdots \oplus W_r$ of subspaces $W_1,\cdots ,W_r$ so that for all $v\in V$ then

$$v=w_1+\cdots +w_r\text{ with }{w}_i\in W_i$$

Let the ${\mathcal{B}}_i$ be a basis for $W_i$ so

$$\mathcal{B}=\bigcup _i{\mathcal{B}}_i\text{ is a basis for }V$$