Thread: Finding Basis'

"Working over the real numbers

R, let U denote the solution set of the system of equations
3x1 + x2 + 7x3 + 4x4 = 0
5x1 + 6x2 + 7x3 + 3x4 = 0
and let W denote the solution set to the system of equations
6x1 + 4x2 + 3x3 + 6x4 = 0
4x1 − x2 + 3x3 + 7x4 = 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

We have



Now, transform the system into row echelon form etc.