Basic Linear Algebra Question I am trying to figure out why
dim(colA) = dim(row A), where
col A is the column space of A and 
row A is the row space of A.
Could someone please help me!
Thank you!!
 A: Denote $\rho^c$ the column rank and $\rho^r$ the row rank.
First of all, $\rho^c(A)=1 \iff A=u\cdot v^T$ for column vectors $u,v$ $\iff \rho^r(A)=1$.
Then prove that $\rho^c(A_1+\ldots+A_n) \le \rho^c(A_1)+\ldots+\rho^c(A_n)$. Finally, for an arbitrary $A$, $r:=\rho^c(A)$, using the $r$ independent columns, construct $r$ pieces of rank 1 matrices $A_i$ such that $A=A_1+\ldots+A_r$.
This shows that $\rho^c(A) = \min\{k \mid \exists A_1,..,A_k$ with rank 1$: A=A_1+\ldots+A_k \}$. And the same for $\rho^r$.
A: Consider $rref(A)$. The column correspondence principle (CCP) states that any linear dependence of the columns of $A$ is also found in the columns of $rref(A)$. It follows that the pivot columns of $A$ form a basis for $Col(A)$.
Next, you can argue that the pivot rows of $rref(A)$ form a basis for the row space of $A$. 
Or, alternatively, argue that the pivot columns of $A^T$ form a basis for $Col(A^T)$ and $Row(A)=Col(A^T)$.
If you haven't proven any of these statements yet then you really should try since these give you simple computations to find bases for row and column space.
