Question

Let U be the subspace of R5 denoted by U =  (x1; x2; x3; x4; x5) in R5 : x1 = 3x2 and x3 = 7x4 : (a) Find a basis of U. (b) Find a subspace W of R5 such that R5 = U $\bigoplus$ W.

Answer 1

a)

Basis of U:

$V_1=(3,1,0,0,0),V_2=(0,0,7,1,0),V_3=(0,0,0,0,1)$

V₁, V₂, V₃ are in U, and are linearly independent.

$u = u_2V_1 + u_4V_2 + u_5V_3$

shows that (V₁, V₂, V₃) spans U.

b)

$V_4=(1,0,0,0,0)$

$V_5=(0,0,1,0,0)$

$W=span(V_4,V_5)$