Question

How do I find a basis of U if U = {(a, b, c) | a + b + c = 0} consider U is a subspace of R^3?

Answer 1

The condition for the coordinates a+b+c=0 means that you have to be on the plane x+y+z=0. So U is a two-dimensional subspace in R^3 therefore it has two basis vectors. For example
\[\textbf{e}_1(1,-1,0)\textrm{ and }\textbf{e}_2(0,-1,1).\]