[Math] Finding set of vectors that spans the solution set

linear algebravector-spaces

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

$$\begin{align}w – x + y + z = 0 \\
5w + 2x – y + z = 0\end{align}$$

Reducing these I get:

When I reduce these, I'm getting

$x = -2z – 6w$

$y = -3z – 7w$

$z = u$

$w = v$

I am getting this as my answer:

$$(x,y,z,w) = u(-2,-3,1,0) + v(-6,-7,0,1)$$

But this is wrong for some reason and I don't understand what I did wrong. Could anyone help me solve this problem correctly?

EDIT: So apparently, the y and z values are correct, however I have my x and w values wrong somehow. According to my instructor, the first entry for u should be -1/7, however I'm not sure how to get that.

Best Answer

I'll advance based on your calculations.

$$ (x,y,z,w) = ( -2u-6v, -3u-7v,u,v ) = ( -2u,-3u,u,0 )+ ( -6v,-7v,0,v )$$

$$\implies (x,y,z,w) = u( -2,-3,1,0 ) + v( -6,-7,0,1 ). $$

Can you see your basis now?

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