Q:

How to decide if a set is a spanning set of a vector space?

I'm having a bit of trouble tackling a problem the algebra teacher gave me. This is the problem:
Let $B= \{ v_1,v_2,v_3,v_4 \} $ be a basis of $\mathbb V$. Decide whether or not $B' = \{v_1-2v_2+v_3,2v_1+v_2+v_4,-v_2+v_3,v_1+v_2+v_3+v_4,5v_1-v_2-v_3-v_4\}$ is a spanning set of $\mathbb V$. 
Any help appreciated!

A:

We need to check whether among the vectors of the set $B'$ we can find $4$ linearly independent vectors, then indicate those vectors as

$w_1=v_1-2v_2+v_3$
$w_2=2v_1+v_2+v_4$
$w_3=-v_2+v_3$
$w_4=v_1+v_2+v_3+v_4$
$w_5=5v_1-v_2-v_3-v_4$

in the basis $B$ they are expressed by

$w_1=(1,-2,1,0)$
$w_2=(2,1,0,1)$
$w_3=(0,-1,1,0)$
$w_4=(1,1,1,1)$
$w_5=(5,-1,-1,-1)$

thus consider the matrix which have those vectors as rows (or columns) and perform the RREF.

