# Row Rank and Column vectors of Matrix

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.

• 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 $$c_j r_j = \begin{pmatrix}4 & 4 \\ 4 & 4\end{pmatrix}$$
– user169852
Nov 7 '20 at 2:58
• i think i meant any column vectors cj, sorry, youre right Nov 7 '20 at 2:59
• Ah, that sounds more plausible!
– user169852
Nov 7 '20 at 2:59

## 1 Answer

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?