How many free variables in this 3x3 matrix without a value for x1?

Question

I am calculating the eigenspace for a matrix that I have now in reduced row echelon form. It looks like this:

$$ \begin{matrix} 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} $$

How many free variables are in this matirx? I'm thinking only 1, x3. But what about the two rows of zeros in the bottom? What about x1?

Answer 1

No, both $x_1$ and $x_3$ are free variables. We have

$x_1=x_1$

$x_2= \quad x_3$

$x_3= \quad x_3$.

With $t=x_1$ and $s=x_3$ we get

$$(x_1,x_2,x_3)^T=t(1,0,0)+s(0,1,1)^T.$$