Let W be the subspace of R3 spanned by the vector w = (2, -3, 1).
Find the basis for W⊥
Describe W⊥ geometrically.
Edit
I have a weak understanding of linear algebra. From what I know a basis is a linearly independent vector(s) that span a subspace. Usually I am given a matrix to orthogonalize using the Gram-Schmidt process then I find the basis by putting that in RREF. I'm not sure where to begin with this problem with one vector
Hint: If $\vec{v}$ is in $W_{\perp}$, then $\vec{v} \cdot \vec{w} = 0$.
Call $\vec{v} = (a,b,c)$. Applying the equation above will give you one equation with three unknowns. Pick values for two of the unknowns and solve for the third. That's one of the basis vectors.
The second basis vector ($\vec{u}$, say) will need to be linearly independent from $\vec{v}$, and perpendicular to $\vec{w}$. (You can pick two different values for the unknowns, solve for the third, and ensure that $\vec{u}$ is not a scalar multiple of $\vec{v}$.)
If it's an orthogonal basis, then $\vec{u}$ will also need to be perpendicular to $\vec{v}$.