Dealing with questions on vectors and matrices with variables I'm running into a lot of questions on linear algebra lately where, instead of using constant numbers in matrices and vectors (these are questions which I have no problems in answering), they use a mix of constants and a variable a. I'm finding these difficult to answer. For example:
For each value of a, find the rank and nullity of the matrix, v1=(a,a,a), v2=(0,1,2) v3=(1,0,-1). I understand rank and nullity but again, I'm thrown off by what is being asked regarding a. If a=0 then the rank is 2 and nullity is 1, correct? Then if a>0, we need to perform additional row operations to reduce to row echelon form. But this can't be done with a variable? I'm mostly having trouble with how I could express the answer correctly considering the inclusions of the variable a. I'm only used to answering LA questions on problems with constants so this has thrown me off.
In general, how can I deal with questions of this type? 
 A: Let's say we are interested in the matrix whose rows are $v_1, v_2, v_3$ (it doesn't really matter as $\mathrm{rank} A = \mathrm{rank} A^T$ so we could as well put them as columns). Thus,
$$ A = \begin{pmatrix} a & a & a \\ 0 & 1 & 2 \\ 1 & 0 & -1 \end{pmatrix}. $$
Instead of handling immediately the case where $a \neq 0$ and $a = 0$, I suggest that you delay it by replacing the third row with the first row, leading to
$$ A' = \begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ a & a & a \end{pmatrix}. $$
By subtracting $a \times $ the first row from the third row, we obtain
$$ A'' =  \begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & a & 2a \end{pmatrix}.$$
Continuing with the second row, we subtract $a \times $ the second row from the third row and obtain
$$ A '' = \begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{pmatrix}. $$
Now it is clear that $A''$ has rank two no matter what $a$ is. In general, you won't be able to avoid case analysis but you can delay it so that expressions involving the parameter will appear only in the last rows which will make the analysis easier.
