What is the basis for subspace: $W=\{x \in \mathbb{R}^4|x_3=x_1+x_2,x_4=x_1-x_2\}$?
I previously posted a similar question regarding showing whether this is a subspace but now I wish to find what the basis is.
I know if we have a linear combination of linearly independent vectors we can have the basis.
So since
$$X=<x_1,x_2,x_1+x_2,x_1-x_2>=x_1<1,0,1,1>+\ x_2<0,1,1,-1>,$$
would the basis be $\{<1,0,1,1>,<0,1,1,-1>\}$ with dimension of subspace $W$ being $2$? Or would it be $4$?
Best Answer
Your work is correct but incomplete. You should show that those two vectors, $\langle 1, 0, 1, 1 \rangle$ and $\langle 0, 1, 1, -1 \rangle$ are linearly independent, as you have only shown that they span $W$. This is not too hard, as they have zeroes in different positions.
Indeed, let $a\langle 1, 0, 1, 1 \rangle + b\langle 0, 1, 1, -1 \rangle = 0$. Then
$$0 = \langle a, 0, a, a \rangle + \langle 0, b, b, -b \rangle = \langle a, b, a + b, a - b \rangle,$$
so $a = b = 0$ and they are linearly independent, and so $\dim W = 2$. Also, I'd like to point out that saying the basis is bad usage. There will be many many bases for $W$. For example, scale up both vectors by $2$. It would be more proper to say a basis.