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For Z, I set the equations as -8x1 - 8x2, -x1 - 2x2, -2x1 - 2x2, and set x1 = 1 and x2 = 1. This gave me the vector [-16,-3,-5] which was the correct answer for a vector that does not span R3 (but I don't know why). I tried setting x1 = 1 and x2 = 2 for W, but it didn't work.

EDIT: I found that if I row reduced the vectors U and V, I got
[1,0
0,1
0,0]
Then I set it as = x3, x2 = x3, x3 = x3. And then x3 = 1 and got the vector [1,1,1] which was the correct answer.
Can someone explain to me why the answers the worked the way they did?

EDIT 2: The image didn't upload for some reason, but the question is below. enter image description here

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    I found no equations on your first statement? – 2017-01-23
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    Sorry, but you have not explained nearly enough. What was $Z$, for instance? And by all means, you should spend a little time learning enough TeX to make your question readable. – 2017-01-23
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    What do you mean by "correct answer for a vector that does not span $\mathbb{R}^3$"? To span $\mathbb{R}^3$ you need at least three vectors - so no single vector spans $\mathbb{R}^3$... – 2017-01-23

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