MHB Finding a set of vectors that span u,v....

[shamieh]

Find a set of vectors {u, v} in $\mathbb{R}^4$ that spans the solution set of the equations:

$x - y + 2z - 2w = 0$

$2x + 2y -z + 3w = 0$

($u$ and $v$ are both $4 \times 1$)
$u = ?$, $v = ?$

I put the matrix in RREF to get

$\begin{bmatrix}1&0&3/4&-1/4\\0&1&-5/4&7/4\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix}$

Then I got $x = -\frac{3}{4} z + \frac{1}{4} w$ and $y = \frac{5}{4} z - \frac{7}{4} w$

But I'm not sure how to present the answer as they want it.

 

[Deveno]

You have what are called "two free variables" (in this case, $z$ and $w$, which must be known to even calculate what $x$ and $y$ are).

Let's write the solutions in a slightly different way, we have:

$4x = -3z + w$
$4y = 5z - 7w$

so we can write $4(x,y,z,w) = (4x,4y,4z,4w) = (-3z+w,5z-7w,4z,4w) = z(-3,5,4,0) + w(1,-7,0,4)$

Maybe this will give you a hint.