# Orthonormal Basis

1. Jul 18, 2009

1. The problem statement, all variables and given/known data

True/False:

The set of vectors $$B={(-1,-1,1,1),(1,0,0,0),(0,1,0,0),(-1,-1,1,-1)}$$ is an orthonormal basis for Euclidean 4-space $$\mathbb{R}^4$$.

2. Relevant equations
None

3. The attempt at a solution

I said false because $$\langle (-1,-1,1,1),(-1,-1,1,1) \rangle =2\ne1$$, which shows that at least one vector in this set is not a unit vector.

However, I'm not sure if I'm supposed to use the usual definition for the inner product. Is this implied by the word "Euclidean"?

2. Jul 18, 2009

### Dick

That looks right. Some of the vectors aren't orthogonal either. "Euclidean" would imply the usual inner product. But even if they left the word "Euclidean" off, I would still use the usual inner product, just because they didn't tell you to use a different one.

3. Jul 18, 2009