How to show the direct sum of two matrices?

[looper]

Homework Statement

Let k be a field, V = Mat2x2(k), U:={[a, b], [-b, a] a, b E k} and W:={[a, b], [b, -a] a, b E k}. Show that V is the direct sum of U and W.

Homework Equations

The Attempt at a Solution

Add the matrix for U to the matrix of W. Values in that matrix still exist in k, so V = U + W. Is that the right reasoning? How do you show the intersection of two matrices is 0? Very confused.

 

[Dick]

Write W={[c, d], [d, -c] c, d E k} to make it clear there are four independent parameters here. Equate an element of U with an element of W and show a=0, b=0, c=0 and d=0 from the equations you get. Then you have to show any element of V can be written as a sum of an element from U and an element from W.