Row Rank and Column vectors of Matrix

Mathematics Asked by james black on December 7, 2020

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.

One 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?

Answered by Franklin Pezzuti Dyer on December 7, 2020

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