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For matrix $A \in M_{m×n}$, let the rank be k and $r_1 , . . . , r_k$ be a subset of rows of A forming a basis of the row-space. How can we show that there exists column vectors $c_j$ such that $A =\sum^k_{j=1} c_j r_j$?

I know that $c_j r_j$ is a mxn matrix and we are adding up a series of $c_j r_j$ together. But I have little idea on how to show there are column vectors $c_j$. Any help or hint is appreciated.

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    $\begingroup$ Are you sure this is true? How would one decompose, say, $$A = \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix}$$ in this way? Pick either row and either column, and you get $$c_j r_j = \begin{pmatrix}4 & 4 \\ 4 & 4\end{pmatrix}$$ $\endgroup$
    – user169852
    Nov 7 '20 at 2:58
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    $\begingroup$ i think i meant any column vectors cj, sorry, youre right $\endgroup$ Nov 7 '20 at 2:59
  • $\begingroup$ Ah, that sounds more plausible! $\endgroup$
    – user169852
    Nov 7 '20 at 2:59
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HINT: Let $r_i$ be the ith row of $A$, and let $c_i^*$ be a column vector with $m$ entries consisting of a $1$ in the ith entry and a $0$ everywhere else. Can you show that this is true?

$$A = \sum_{i=0}^m c_i^{*} r_i$$

Once you see why this is true, can you figure out a way to use this fact to solve your problem?

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