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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?

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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 $\Bbb R^n$.

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