How can a rank 1 complex matrix be written as a product of two matrices?

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Here is this week's POTW:

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If $M$ is a complex $m \times n$ matrix of rank $1$, show that $M$ can be written as $\bf{uv^T}$ where $\bf{u}$ is an $m\times 1$ matrix and $\bf{v}$ is an $n\times 1$ matrix.

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Congratulations to Opalg and castor28 for their correct solutions. Here is castor28's solution.

Spoiler

As the column space of $M$ has dimension $1$, it is spanned by a single vector $\mathbf{u}$. Therefore, for all $i$, the column $i$ of $M$ can be written as $\mathbf{u}v_i$ for some scalar $v_i$.

This shows that $M=\mathbf{uv^T}$, where $\mathbf{v^T}$ is the column vector $(v_i)$.