I have an elementary question about linear dependence and column/row spaces.
Suppose we have a matrix $A=\begin{bmatrix}1&2\\2&4\end{bmatrix}$, and we construct the matrix $B=\begin{bmatrix}1&2\\3&6\end{bmatrix}$ as a combination of the rows of $A$.
We are asked whether the columns of $B$ are a combination of the columns of $A$.
Just by looking at the first column $b_1$, it sure seems like there is not going to be any way of representing this as linear combinations of the columns of $A$.
But, I recall that for a symmetric matrix $A=A^\text{T}$, the column space and row space are "equivalent". But $\begin{bmatrix}1\\3\end{bmatrix}\not\in C(A)$!
So there is clearly some form of error in my understanding and reasoning here. Please, may somebody enlighten me? I would guess it's a misunderstanding I have about in what way the column space and row space are related in a symmetric matrix?