03 - Types of Sets

Linearly Independent Set

Let $V$ be a vector space over $\mathbf{F}$

Set $S\subseteq V$ is linearly independent if for

$$a_1,\cdots ,a_n\in \mathbf{F}\text{ and }{s}_1,\cdots ,s_n\in S$$

then

$$a_1{s}_1+\cdots +a_n{s}_n⟹a_1=\cdots =a_n=0$$

Spanning Set

Let $V$ be a vector space over $\mathbf{F}$

Set $S\subseteq V$ is spanning if for all $v\in V$ there exists

$$a_1,\cdots ,a_n\in \mathbf{F}\text{ and }{s}_1,\cdots ,s_n\in S$$

such that

$$v=a_1{s}_1+\cdots +a_n{s}_n$$

Basis Set

Let $V$ be a vector space over $\mathbf{F}$

Set $\mathcal{B}\subseteq V$ is a basis of $V$ if

$$B\text{ is spanning and linearly independent}$$

Dimension of a Vector Space Size of $\mathcal{B}$ is the dimension of $V$