Calculating a basis in $\mathbb{R}^4$.

Question

Have the subspace of $\mathbb{R}^4$

$$W = \{(x,y,z,w) \in\mathbb{R}^4 : y - w + z = 0\}$$

Calculate a basis for $W$, and then find an orthonormal base from that.

The basis, from the book, is

$$\{(1,0,0,0) , (0,1,0,1) , (0,-1,1,0)\}$$

Wait, if we're in $\mathbb{R}^4$, shouldn't the basis have four vectors?

Also, how do you calculate an orthonormal basis out of the given basis above? (I can't find the corresponding explanation).

Answer 1

As $W$ is a subspace of $\mathbb{R}^4$, it could have dimension less than four.

To get an orthonormal basis for a subspace, you can apply the Gram-Schmidt process to any basis for that subspace.