The exercise tells me that given the set
$$W=\{(x_1,x_2,x_3,x_4,x_5) \in \mathbb{R}^5 : x_1+x_2=x_3-x_4 , x_2+2x_3=x_4-2x_5\}$$
I should
- Prove that $W$ is a vector subspace of $\mathbb{R}^5$ (which I think I did good, I just applied the 3 conditions ($W$ different than zero//closed by addition//and closed by scalar multiplication. If that's somehow wrong, please tell me, because my teacher has done it in a totally different and weird way)
And the hard one, the one that's making me ask this
- Give the parametric expression of the subspace. I think maybe the kartesian equation would work here, but I can't figure out how, I only have one resolved example on $\mathbb{R}^4$. So if any of you guys could help me, I can't thank you enough.
Since you have 2 constraints put on top of $\mathbb R^5$ this sub-space should have dimension 3.
$x_1 =s\\ x_2 = t\\ x_3 = u$
From the first constraint $x_4 = -s - t + u$
And plug these into the second constraint
$x_5 = \frac 12 x_4 - \frac 12 x_2 - x_3\\ x_5 = -\frac 12 s - \frac 12 t + \frac 12 u - \frac 12 t - u\\ x_5 = -\frac 12 s - t - \frac 12 u $
$W = \pmatrix{1\\0\\0\\-1\\-\frac 12}s + \pmatrix{0\\1\\0\\-1\\-1}t +\pmatrix{0\\0\\1\\1\\-\frac 12}u$