find equality between linear spans

Question

$$U = Sp\{(2,5,-4,-10), (1,1,1,1),(1,0,3,5), (0,2,-4,-8)\}$$ $$ W = Sp\{(1,-2,7,13), (3,1,7,11), (2,1,4,6) \}$$

two questions:

1. prove that $U = W$
2. find the values of the $a \in \mathbb{R}$ where the vector $v=(a,a-6,4a-3,6a-1)$ belongs to $U$

i thought about solving it with a matrix but i wasn't sure how to approach it maybe something like

$$ \begin{pmatrix} 2 & 1 & 1 & 0 &| & 1 & 3 & 2 \\ 5 & 1 & 0 & 2 &| & -2 &1 &1\\ -4 & 1 & 3 & -4&|&7& 7& 4 \\ -10 & 1 & 5 & -8&| & 13 &11 &6 \end{pmatrix} $$

and to use row reduction some how to reach zero maybe?

also what about the second question was lost there also

Answer 1

Let \begin{align*} A &= \begin{bmatrix} 2&1&1&0\\ 5&1&0&2\\ -4&1&3&-4\\ -10&1&5&-8 \end{bmatrix} & B&= \begin{bmatrix} 1&3&2\\ -2&1&1\\ 7&7&4\\ 13&11&6 \end{bmatrix} \end{align*} Note the equation $AX=B$ has a (non-unique!) solution $$ X= \begin{bmatrix} -1&-2/3&-1/3\\ 3&13/3&8/3\\ 0&0&0\\ 0&0&0 \end{bmatrix} $$ Also note the equation $BY=A$ has a (also non-unique!) solution $$ Y= \begin{bmatrix} -13/7&-2/7&1/7&-6/7\\ 9/7&3/7&2/7&2/7\\ 0&0&0&0 \end{bmatrix} $$ Do you see how these equations relate to your problem?

That $AX=B$ has a solution implies that $U\subseteq W$ and that $BY=A$ has a solution implies that $W\subseteq U$. Hence $U=W$.