Let U be the subspace of F4 given by U x x y y x y elemento

Let U be the subspace of F^4 given by U = {(x, x, y, y): x, y elementof F}. (a) Find a basis for U. (b) Find a basis for F^4 which contains your answer to part (a) as a sublist.

Solution

(a). An arbitrary element of U is X = (x,x,y,y) = x(1,1,0,0)+y(0,0,1,1). Hence a basis for U is {(1,1,0,0),(0,0,1,1)}.

(b) The required basis for F⁴ is S = {(1,1,0,0),(0,0,1,1),(0,1,1,0), (1,0,1,0)}

Note: The vRREF of the matrix with the vectors in S is I₄. Hence these vectors are linearly independent and span F⁴.