Three subspaces of R^4×1; U1 =[a], U2 =[{b,c}], U3 =[{b, a}]
a = (0, 1, 1, 0)^T , b = (1, 0, 0, 1)^T , c = (0, 1, 0, 0)^T.
Determine bases of U1 + U2, U2 + U3, U1 + U2 + U3. Which of these sums are direct?
I managed to do U1+U2;
U1+U2={a1*(0,1,1,0) + a2*(1,0,0,1) + a3*(0,1,0,0)}
The basis is {(0, 1, 1, 0)^T ,(1, 0, 0, 1)^T, (0, 1, 0, 0)^T)}
And the sum is direct because U1 ∩ U2 is the null vector.
For U2+U3 I got:
U2+U3= {a2*(1,0,0,1) a3*(0,1,0,0)+ a2*(1,0,0,1) + a1*(0,1,1,0)}
I don't know how to find a basis and if the sum is direct.
For U1+U2+U3 I can see that the sum isn't direct but again can't find a basis if there even is one.