Do the given vectors span $\mathbb{R}^3$?

Question

Do the following vectors span $\mathbb{R}^3$:

$$v_1 = (2, -1,3)$$ $$v_2 = (4, 1, 2)$$ $$v_3 = (8, -1, 8)$$

I use Gaussian Elimination to bring the matrix to an echelon form, with a pivot of "1" in every row. I conclude that the vectors span because of the above reason, however the answer of the problem is "the vectors do not span". Am I doing something wrong?

Answer 1

Note that $v_3 = 2v_1 + v_2$. This means they are not linearly independent, so they span a subspace of dimension at most (and equal to in this case) $3-1=2$, not the whole space. You probably got something wrong in the calculations or have misinterpreted a row of all zeros.