Understanding Matousek's proof of Equiangular lines In Miniature 9 in 33 Miniatures by Matousek, he proofs that:

The largest number of equiangular lines in $\mathbb R^3$ is 6, and in general, there cannot be more than $\binom{d+1}{2}$ equiangular lines in $\mathbb R^d$.

The proof starts with:
Let us consider a configuration of $n$ lines, where each pair has the same angle $\vartheta \in (0, \pi/2]$. Let $v_i$ be a unit vector in the direction of the $i$th line (we choose one of the two possible orientations of $v_i$ arbitrarily). The condition of equal angles is equivalent to
$$ |\langle v_i, v_j\rangle| = \cos \vartheta, \quad \text{for all } i\neq j.$$
Let us regard $v_i$ as a column vector, or a $d\times 1$ matrix. Then $v_i^Tv_j$ is the scalar product $\langle v_i, v_j \rangle$. On the other hand, $v_iv_j^T$ is a $d\times d$ matrix.
We show that the matrices $v_iv_i^T, i = 1,2,\dots, n$ are linearly independent.
And I do not see how showing the independence of these matrices shows that we can not have more than the claimed number of equiangular lines?
Thanks!

The key may be the first sentence "Let us consider a configuration of  lines..."
You can think of it as proof by contradiction. If we had more than 6 equiangular lines, we would construct more than 6 such independent symmetric $3\times 3$ matrices (by the outlined procedure), which is impossible.
