Let $f$ be a symmetric endomorphism of $\mathbf{R}^n$ with eigenvalues $\lambda_1 \leqslant \ldots \leqslant \lambda_n$ and associated orthonormal eigenbasis $(e_1, \ldots, e_n)$.
Let $k \in \llbracket 1, n \rrbracket$ and $S_k$ a vector subspace of $\mathbf{R}^n$ of dimension $k$. We set $T_k = \operatorname{Vect}(e_k, \ldots, e_n)$.
Justify that $S_k \cap T_k \neq \{0\}$.