Span

The span of a vector list is the set of all its real linear combinations.

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01The span of supplied vectors contains every combination made with real coefficients#

The span of supplied vectors contains every combination made with real coefficients. It includes multiples, sums and zero.

Use $u=(1,0)$ and $v=(1,2)$ . Test $au+cv=(4,6)$ .

Match coordinates.

\begin{gathered}a+c=4\\2c=6\end{gathered}

$c=3$ , then $a=1$ . Both coordinates match, so the target is in the span.

For $u=(1,0)$ and $v=(2,0)$ , every combination is $(a+2c,0)$ . The target $(3,1)$ would require $0=1$ in the second coordinate.

The coefficient system must match every target coordinate. Its consistency is exactly the test for span membership.

Check your reasoning

To reach $(5,4)$ , solve $a(1,0)+c(1,2)=(5,4)$ . What is $a-c$ ?

Show answer and explanation

$2c=4$ gives $c=2$ ; $a+c=5$ gives $a=3$ .

Using $u=(1,0)$ and $v=(2,0)$ , both $(a,c)=(2,0)$ and $(0,1)$ produce $(2,0)$ .

One working choice proves membership. The coefficients need not be unique, nonnegative, or add up to one.

Check your reasoning

Per unit, A adds $(1,0)$ and B adds $(0,2)$ . Can signed real settings produce $(-2,6)$ ?

Show answer and explanation

Yes: A= $-2$ , B= $3$ .

$(-2,2\cdot3)=(-2,6)$ .

Check your reasoning

$u=(1,2)$ . Rae excludes $(0,0)$ from its span. Refute this.

Show answer and explanation

$0u=(0,0)$ .

Use coefficient 0.

Key takeaway