1. ## constructing basis

Define a real linear transformation
L1 : R4 \to R2 by

L
1(x1, x2, x3, x4) = (3x1 + x2 + 2x3 x4, 2x1 + 4x2 + 5x3 x4)

and let
U1 denote the kernel of L1. Define a real linear transformation L2 : R4 ! R2 by

L
2(x1, x2, x3, x4) = (5x1 + 7x2 + 11x3 + 3x4, 2x1 + 6x2 + 9x3 + 4x4)

and let
U2 denote the kernel of L2.

Construct bases for U1, U2, U1 &U2 and U1 + U2.

Not sure on how to tackle this question. Or how to construct basis in similar questions. Any help would be appreciated a lot!

2. The kernel of a linear transformation, L, is, by definition, the space of vectors, v, such that L(v)= 0. If $(x_1, x_2, x_3, x_4)$ is in the kernel of $L_1$ then $3x_1 + x_2 + 2x_3- x_4= 0$, $2x_1 + 4x_2 + 5x_3- x_4= 0$.

We can solve those two equations for two of the unknowns in terms of the other two. For example If we subtract the second equation from the first, we eliminate $x_4$: $x_1- 3x_2- 3x_3= 0$ so that $x_1= 3x_2+ 3x_3$. Putting that back into the second equation, [tex]2(3x_2+ 3x_3)+ 4x_2+ 5x_3- x_4= 10x_2+ 11x_3- x_4[tex] so that $x_4= 10x_2+ 11x_3$.

That is, we can write $(x_1, x_2, x_3)= (3x_2+ 3x_3, x_2, x_3, 10x_2+ 11x_3)= (3x_2, x_2, 10x_2)+ (3x_3, 0, x_3, 11x_3)= x_2(3, 1, 0, 10)+ x_3(3, 0, 1, 11)$

Now, do you see what a basis of $U_1$ is?

Do the same thing for $U_2$.

To find a basis for $U_1+ U_2$ (I assume you mean the direct sum), combine the two bases, then throw out any that can be written as a linear combination of others. I don't know what " $U_1$& $U_2$" means.

3. by U1&U2, he probably means the intersection.

by U1+U2, he probably means the sum (not the direct sum).

these would be the meet and join of the subspaces in the lattice of subspaces of R^4.

4. Thanks that helped.

So the basis of U_1 is : {(3,1,0,10),(3,0,1,11)}

And yes sorry i meant the intersection.

So the basis of U_1 + U_2 , is the combined basis, but then I should apply the pruning process to get rid of some?

How do i get the basis of intersection of U_1 U_2 ?

5. Im having problems finding the basis from U2. Is there not a process i should be using like matrices, rather than just doing it by inspection.