Finding Basis'

• May 13th 2011, 09:37 AM
sj247
Finding Basis'
"Working over the real numbers
R, let U denote the solution set of the system of equations
3
x1 + x2 + 7x3 + 4x4 = 0
5
x1 + 6x2 + 7x3 + 3x4 = 0
and let
W denote the solution set to the system of equations
6
x1 + 4x2 + 3x3 + 6x4 = 0
4
x1 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(Rofl)

• May 13th 2011, 09:05 PM
FernandoRevilla
We have

$\displaystyle U\cap V\equiv \begin{Bmatrix}3x_1+x_2+7x_3+4x_4=0\\ 5x_1+6x_2+7x_3+3x_4=0\\6x_1+4x_2+3x_3+6x_4=0\\4x_1-x_2+3x_3+7x_4=0\end{matrix}$

Now, transform the system into row echelon form etc.