I'm struggling to find a solution this question, which involves a variable. I need to find the basis of the span of the following vectors: $(a,1,1), (1,a,1)$ and $(a,1,1)$ for all values of $a$. Obviously $v_1$ and $v_3$ are the same vectors for all values of $a$ so I'm left with two vectors $(a,1,1)$ and $(1,a,1)$. I understand how to find the basis of a span of vectors when dealing with all constants so this is what I have so far:
$x(a,1,1) + b(1,a,1) = (0,0,0)$
$(ax + b, x+ba, x+b) = (0,0,0)$
$x=-b$
So $(-ba + b, -b+ba, -b+b) = (0,0,0)$
$-ba+b=0$
$-ba=-b$
$a=1$
I don't even know if this is relevant and if it answers the question. What do I need to do to show a basis of a span for all values of $a$ sufficiently?
Put the vectors as rows of the matrix and do the row reduction to get
$\pmatrix{ a & 1 & 1 \\ 1 & a & 1 \\}\rightarrow \pmatrix{ 1 & a & 1 \\ a & 1 & 1 \\}\rightarrow \pmatrix{ 1 & a & 1 \\ 0 & 1-a^2 & 1-a \\}\rightarrow \pmatrix{ 1 & a & 1 \\ 0 & 1 & \frac1{1+a} \\}\rightarrow \pmatrix{ 1 & 0 & \frac{1}{a+1} \\ 0 & 1 & \frac{1}{a+1} \\ }$
conclude that for $a\ne1$ the vectors are linearly independent and therefore form a basis for a given subspace. For $| a|\ne1$ the rows of the final matrix are the basis you looking for. For $a=-1$ the basis is $(1,-1,1),(0,0,1)$ and for the case $a=1$ the basis is $(1,1,1)$.