Which set Spans the Same set

Question

Which set spans the same set as $\left\lbrace(1,2,−1),(0,1,1),(2,5,−1)\right\rbrace$ ?

The answers choices each give either a set of 3 vectors or 2 vectors. How will I go about to solve this question?

Answer 1

Take just the first two vectors. The last is $2v_{1}+v_{2}$.

Answer 2

A foolproof way to show that $A$ and $B$ span the same space is to show that each vector in $A$ can be made via combinations of vectors from $B$ and vice versa. This is essentially just setting up some simultaneous linear equations and solving them, for which there exist a variety of methods.

You can make this easier by first altering $A$ or $B$ (or both) to make a set with the same span but "simpler" elements. In this case, as the other answer points out, you can discard the third vector, since you can make it with the first two. You can also combine the vectors in different ways to make them simpler, e.g. you could replace the first vector with, say, $v_1 + v_2 = (1,3,0)$, since you can get back to the original first vector by subtracting the second. The zero in the third component is especially useful: now if $(x,y,z)$ is going to be a combination of $v_1 = (1,3,0)$ and $v_2 = (0,1,1)$, you can immediately deduce it must be $xv_1 + zv_2$ for the first and third coefficients to match, so you just need to check if the second does too.