# Rank of sum of dependent outer product matrices

I have $K$ column vectors $u_i$ of which only $K-1$ are linearly independent. A matrix $S$ is calculated from a linear combination of outer products of these vectors with themselves:

$S=\Sigma (a_i(u_i)(u_i)')\space\space i=1,...,K$

I know $rank(S) \le K-1$, but why?

The rank of a matrix is the dimension of its image. In this case, the image lies within the span of the vectors $u_i$, and this span is a $k-1$ dimensional subspace of $\Bbb R^n$.