Let

v1 = (1, 0, −1)

v2 = (0, 2, 2)

v3 = (−3, 4, 7)

and let W = Span{v1, v2, v3}.

What is the dimension of W? Is it 2?

Any explanation would be great too.
Thanks.

Yes because you can make V3 out of a linear combination of V1 and V2

V3=-3V1+2V2

that means that all three lie in a two dimensional plane. We could express all three as a linear combination of two orthogonal vectors in that plane.