How do I show that W is an orthogonal basis?

Question

Given that subspace $W = \text{span}\{[1 1 0 -1], [1 0 1 1], [0 -1 1 -1]\}$, how do I show that $W$ (not subspace $W$) is an orthogonal basis?

Any help would be greatly appreciated.

Answer 1

Every family of orthogonal vectors is linearly independent- I think intuitively this makes sense by the definition of orthogonality.

So, it is sufficient to check orthogonality. Simply apply the inner product on your space pairwise, in this case the dot product for $\mathbb{R}^4$. If the value it returns is zero, then you know that the vectors are orthogonal.

Answer 2

Show that any vector in your space is a combination of your vectors in W, and then show that your vectors in W are orthogonal.