Quote by lkh1986
For question 1 with the condition x + y + 2z = 0, try to express one of the variables in other two variables. For example, x = y  2z. Then, replace x with y  2z in {(x,y,z)x,y,z are real}. Lastly, try to "factor" or decompose the vector into several vectors to get the basis.

Alright, I'll give it a shot.
x = y 2z
{(y,z)  y2z, y, z}
{(y,z)  y(1, 1, 0) + z(2, 0, 1)}
Basis: {(1, 1, 0), (2, 0, 1)}
Is that correct?