Suppose $v_{1}, ..., v_{k+1}$ is a linearly independent, orthonormal set of vectors in $R^{n}$. Prove that for $Y \in R^{n \times k}$ there exists a non-trivial linear combination of these vectors $w$ such that $Y^{T}w = 0$.
I was looking into Proof of Eckart–Young–Mirsky theorem (for spectral norm) in wiki https://en.wikipedia.org/wiki/Low-rank_approximation and couldn't understand this argument. I would appreciate any help with this.