I have a non-complete set of orthogonal vectors $V=[\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n]$ with $m > n$ entries. I would like to choose another vector $\mathbf{w}$ which is linearly independent from $\mathbf{v}_i$ for all $1 \le i \le n$. What is the safest and most stable way to choose such $\mathbf{w}$? If the entries of $V$ are approximate eigenvectors of an $m \times m$ matrix $\mathbf{A}$, then would $\mathbf{w}=\mathbf{A}(\mathbf{v}_1+\mathbf{v}_2+\ldots+\mathbf{v}_n)$ be a good choice?
2026-03-28 16:26:23.1774715183
How to choose a vector which is linearly independent from a set of orthogonal vectors?
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