Suppose that $A_{mn}$ is a matrix over some field, and that $C, R$ is its column space and row space, without using the fact that $rank(C) = rank(R)$, can we show that, there exists a bijection between $C$ and $R$?
[Math] Bijection between column space and row space
linear algebra
Best Answer
Hint: For $x\in R$, consider $Ax\in C$. Show this map is a bijection. (It might be useful to note that $\Bbb R^n=R\oplus N$, where $N=\{x: Ax=0\}$.)