"Working over the real numbers

**R**, let *U *denote the solution set of the system of equations

3*x*1 + *x*2 + 7*x*3 + 4*x*4 = 0

5*x*1 + 6*x*2 + 7*x*3 + 3*x*4 = 0

and let *W *denote the solution set to the system of equations

6*x*1 + 4*x*2 + 3*x*3 + 6*x*4 = 0

4*x*1 *− x*2 + 3*x*3 + 7*x*4 = 0*.*

Construct bases for

*U*, *W*, *U ∩W *and *U *+*W*.

"

Im think i'm doing this right, but not 100% sure.

To find basis for U:

Write solution set in augmented coefficent form

Get in row reduced echleon form

Write two of (x1, x2, x3, x4) in terms of other two. Then re-write in

(x1, x2, x3, x4) form.

Then seperate the two terms like(not from example) "x2( 3, 0, 1, 11) + x3( 4, 7, 0 9)".

Which givesthe basis (3, 0, 1, 11) & (4, 7, 0 9).

Do the same for W

For U+ W, add basis together then apply a pruning process.

For U intersection W, I don't have a clue!

Would be very grateful for any help