The kernel of a linear transformation, L, is, by definition, the space of vectors, v, such that L(v)= 0. If is in the kernel of then , .

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 : so that . 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 .

That is, we can write

Now, do you see what a basis of is?

Do the same thing for .

To find a basis for (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 " & " means.