Find a basis of subspaces U and W

Question

U = {(x1, x2, x3, x4) ∈ R 4 | x2 − x4 = 0 ∧ x1 = 0} ;

W = {(x1, x2, x3, x4) ∈ R 4 | x2 = 0 = x3} ;

Find a basis of subspaces U and W. Are the 2 subspaces complement to one another?

Find a base of orthogonal complement U*

Answer 1

Here is some help with the first part:

For subspace U, doing some rearranging leads to the conditions that x1=0 and x2=x4, where x1,x2,x3,x4 are real numbers. Parametrizing, x1=0, x2=s (s∈R), x3=t (t∈R), x4=s (since x2=x4 from above).

Then any vector v in U can be represented as v=(0, s, t, s), or as a linear combination of basis vectors b1 and b2:

v=s*b1+t*b2=(0, s, t, s)=s(0, 1, 0, 1)+t(0, 0, 1, 0)

Then the set of basis vectors B_u∈U is B_u={(0, 1, 0, 1), (0, 0, 1, 0)}

For W, a similar process yields x1=s (s∈R), x2=0, x3=0, x4=t (t∈R)

Then any vector v in W can be represented as v=(s, 0, 0, t), or as a linear combination of basis vectors:

v=(s, 0, 0, t)=s(1, 0, 0, 0)+t(0, 0, 0, 1)

Then the set of basis vectors B_w∈W is B_w={(1, 0, 0, 0), (0, 0, 0, 1)}