We've got the following span: $$U = Sp\{(2, 5, -4, -10), (1, 1, 1, 1), (1, 0,3,5) , (0,2,-4,-8)\}$$
We need to find the values of the number $a$ where the vector $$v = (a, a-6, 4a-3, 6a-1)$$ belongs to $U$.
since $U$ is a span I know that I can use it as linear combination
Let $U = Sp\{u1,u2,u3,u4\}$ $$v = a1u1 + a2u2 + a3u3 + a4u4$$
we can use a matrix now:
$$ \begin{bmatrix} 2&1&1&0& |& a \\ 5&1&0&2&|&a-6\\ -4&1&3&-4&|&4a-3\\ -10&1&5&-8&|&6a-1 \end{bmatrix} $$
do I need to use row reduction now? also, when I'll get canonical matrix how do I continue from there also?
Ok so I finally was able to solve it after an hour of attemp's using row reduction!
this is my solution: I used this matrix from the question $$ \begin{bmatrix} 2&1&1&0& |& a \\ 5&1&0&2&|&a-6\\ -4&1&3&-4&|&4a-3\\ -10&1&5&-8&|&6a-1 \end{bmatrix} $$
and done row reduction until I got $$ \begin{bmatrix} 2&1&1&0& |& a \\ 5&1&0&2&|&a-6\\ 0&3&5&-4&|&6a-3\\ 0&0&0&0&|&2a-10 \end{bmatrix} $$
which means that $a$ must be equal to 5 in order for it to belong to $U$.
(any additions to this answer will be much appreciated)