# Finding a set of vectors that spans the solution

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.

• HINT: (Assuming your numbers are correct) you've found that $$\pmatrix{x \\ y \\ z \\ w} = \pmatrix{-\frac 34z+\frac 14w \\ \frac 54z-\frac 74w \\ z \\ w} = z\pmatrix{-\frac 34 \\ \frac 54 \\ 1 \\ 0} + w\pmatrix{\frac 14 \\ -\frac 74 \\ 0 \\1}$$ – user137731 Jul 8 '16 at 3:03

You can write the solution set $S$ as follows: $$S=\{(x,y,z,w)\in\mathbb{R}^4: x=-\frac{3}{4}z+\frac{1}{4}w,\, y=\frac{5}{4}z-\frac{7}{4}w;\,\,\ z,w\in\mathbb{R}\}.$$