Say we have a set of symmetric $n \times n$ matrices $M_i$ for $1 \leq i \leq k$, elements in $\mathbb{R}$. Suppose that for every $\boldsymbol{\lambda} = (\lambda_1, \dots , \lambda_k) \in \mathbb{R}^k$ we have that the kernel of
\begin{equation*} M_{\boldsymbol{\lambda}} = \sum_i \lambda_i M_i \end{equation*}
is nontrivial. Does it follow that there exists some nonzero $n$ vector $\textbf{v}$ with $M_i \textbf{v} = 0$ for all $i$?