Question

Find the dimension and a basis for 𝑊 = {(𝑥, 𝑦, 𝑧, 𝑤,𝑡): 𝑥 + 𝑦 + 𝑧 + 𝑤 + 𝑡 = 0, 𝑥 − 𝑦 + 𝑧 − 𝑤 + 𝑡 = 0}

Answer 1

$W$ is a subspace of $\mathbb{R}^5$. From equations $x+y+z+w+t=0,x-y+z-w+t=0$ we find: $t=w-z+y-x$ . Substitute it in the first equation and get: $x+y+z+w+w-z+y-x=2y+2w=0$ . Thus, we get: $t=-z-x$, $w=-y.$ Thus, the space is 3 dimensional. $x,y,z$ are independent coordinates and coordinates $t$ and $w$ can be expressed via $x,y,z$ . To find the basis we take the standard basis from $\mathbb{R}^3$: $x_1=1,y_1=0,z_1=0$, $x_2=0,y_2=1,z_1=0$ and $x_3=0,y_3=0,z_3=1$ and extend it to $\mathbb{R}^5$ by using equations for $t$ and $w$. We receive 3 vectors: $e_1=(1,0,0,0,-1), e_2=(0,1,0,-1,0)$ and $e_3=(0,0,1,0,-1)$. This is a basis of $W$.