For the subset $Y$ of $\mathbb R^5$ given below, find a spanning set and a basis, and hence determine its dimension.
$Y = \{(x_1, x_2, x_3, x_4, x_5 ) \in \mathbb R^5 |x_1 +x_3 = 0, x_1 −x_2 +x_5 = 0\}.$
First I tried to find the spanning set. I used elimination. I have $x_1 = -x_3$ and $x_2 = -x_3 + x_5$. This gives $Y = \{(-x_3,-x_3+x_5,x_3,x_4,x_5 ) \} = x_1(0,0,0,0,0) + x_2(0,0,0,0,0) + x_3(-1,-1,1,0,0)+x_4(0,0,0,1,0)+x_5(0,1,0,0,1)$.
So, I believe we can write $Y=$span$\{(-1,-1,1,0,0),(0,0,0,1,0),(0,1,0,0,1) \}$.
I already know that $\{(-1,-1,1,0,0),(0,0,0,1,0),(0,1,0,0,1) \}$ is a spanning set for $Y$. Since these three vectors are linearly independent, we must have that $\{(-1,-1,1,0,0),(0,0,0,1,0),(0,1,0,0,1) \}$ is also a basis for $Y$, with dim($Y$)$=3$.