Spanning Set

The the Subset of Vector Space $V$ Spans $V$.

Definition

A subset $S\subset V$ is a spanning set if:

• $V=span(S)$ Look at: Span

Examples

Element of Set

$U=span\{[1,0,0],[0,0,1]\}$ An element of this spanning set could be: $[\mathrm{5.0.9}]$ as $[5,0,9]$ can be written as a Linear Combination of $[1,0,0]$ and $[0,0,1]$

Checking Elementhood Test of a vector

Say we have $U=span\{[1,1,-1],[1,2,-1]\}$ If we want to check if $\overrightarrow{[1,2,3]}\in U$ , we need to find $a,b\in R$ s.t $a\overrightarrow{[1,1,-1]}+b\overrightarrow{[1,2,-1]}=\overrightarrow{[1,2,3]}$ So, we solve:

• $a+b=1$
• $a+2b=2$
• $-a-b=3$ There are no values $a,b$ that satisfy this case.

Concepts

• Non-Colinear Vectors Spanning Set
• Proving Spanning Sets Equivalent
• Span of Empty Set