Correct criterion for the difference between two subspaces

Question

Let us denote $V = sp(v_1,...,v_k)$ where $v \in \mathbb{R}^n$. I have an algorithm that approximates the vector $v_i$ by some $v'_i$. Denote $V' = sp(v'_1,...,v'_k)$. I am interested in the "difference" between $V$ and $V'$, and not the individual vectors. What would be right criterion for this?

Thanks.

Answer 1

One possibility is to use ranks.

Let $A$ be the matrix with the coordinates of a basis of $V$.

Let $B$ be $A$ augmented with the coordinates of a basis of $V'$.

The difference $f(V,V')=\text{rank}(B) - \text{rank}(A)$ measures how far $V'$ is from $V$.

If you want a symmetric function, use $f(V,V')+f(V',V)$.