Conditions for this vectors to be linearly dependent Consider $m<n$ positive definite matrices $A_1\dots,A_m\in\mathbb{R}^{n\times n}$ which are linearly independent matrices. This is, that there are no $c_1,\dots,c_n\in\mathbb{R}$ such that $\sum_{i=1}^m c_iA_i = 0$ unless $c_i=0, \forall i=1,\dots,m$.
How can I find vectors $x\in\mathbb{R}^n$ different from $0$ such that the set of vectors
$$
A_1x,\dots,A_mx
$$
become linearly dependent? This is, that there exists  $d_1,\dots,d_n\in\mathbb{R}$ different from all $0$,such that $\sum_{i=1}^m d_iA_ix = 0$.
This question is motivated by an example as the following. Consider the matrices
$$
A_i = \begin{bmatrix}
M & 0_{3 \times 1}\\
0_{1\times 3} & a_i
\end{bmatrix}
$$
where $a_i\in\mathbb{R}$ and $M\in\mathbb{R}^{3\times 3}$, and $a_1,\dots,a_m$ are different from each other. Hence, if I take
$$
x = \begin{bmatrix}
x' \\
0
\end{bmatrix}
$$
with some $x'\in\mathbb{R}^3$, then $A_ix = Mx', \forall i=1,\dots, m$. Thus, in this case (which is something like a "trivial example") we have that the vectors $A_ix$ are linearly dependent.
 A: Not a full answer, but too long for a comment.
Essentially you are asking if you can find a nontrivial combination of the matrices that is singular. (If you can do that, then the combination $(\sum \alpha_i A_i) x=0$ will have a solution for some $x$.)
This seems to be related: Vector subspace of $M_n(\mathbb{R})$ with invertible matrices
In general, you can not do it for $m<n$.
Without the positive definiteness, a counter-example is trivial: take, in $\mathbb{R}^4$, the identity matrix and the matrix that rotates $x-y$ plane by $90$ degrees and simultaneously rotates the $z-w$ plane by $90$ degrees.
With positive definite symmetric matrices, I did a few experiments in python and here is a counter-example for 2 matrices:
$$
A = \mathrm{diag}(4,3,2,1)
$$
$$
R = \begin{pmatrix}
-2.00850073 & -1.60957843 & -0.90369738 & -0.50794833\\
 2.38498604 &  0.57307146 & -0.07249097 &  1.52387285\\
-0.57167226 &  0.40667149 & -1.85512324 &  0.44869258\\
-0.21978557 & 1.07929074 & -1.82720749 & -1.99403596
\end{pmatrix}
$$
a random matrix, and $B = RAR^{-1}$.
This is the plot of $\det(A + \lambda(B))$ (always positive):

A: We have that
$$d_1A_1x+d_2A_2x+\ldots+d_mA_mx=0 \iff (d_1A_1+d_2A_2+\ldots+d_mA_m)x=0$$
that is $x \in \ker(d_1A_1+d_2A_2+\ldots+d_mA_m)$ then the problem is equivalent to find $d_i$ such that $\operatorname{rank}(d_1A_1+d_2A_2+\ldots+d_mA_m)<n$.
