I am taking my first Linear Algebra Class in college and it is one of the hardest math classes I have ever taken. It is my introduction to proofs and the semester just started. I am very lost in the class and here is an example of one problem I can't seem to understand.
I have to prove that $\mathrm{Col}_i (A') = \mathrm{Row}_i (A)'$ and that $\mathrm{Col}_j (A') = \mathrm{Row}_j (A)'$. In the problem it says "the transpose of $A$ is the matrix $A' = (b_{ij})$ where $b_{ij} = a_{ji}$".
It would be great to have an abstract explanation and an applied explanation, which I think will help me understand it much better.
Thanks in advance.
As I said in the comments (I post this as an answer, since it seems that the problem was solved):
Try to work with little examples, for instance $$A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$ Do you see why $\mathrm{Col}_{1}(A')=\mathrm{Row}_{1}(A)'$ ? Replace $A$ by $$B=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}$$ Do you see why $\mathrm{Col}_{1}(B')=\mathrm{Row}_{1}(B)'$ ?
This is just a way to understand how to get the intuition and how you can see what happens.
For writing your proof, you could start by something like : Let $M$ a $n \times m$ matrix $$M=\begin{bmatrix} a_{11} & a_{12} & \dots & a_{1m}\\ \vdots & \vdots & \vdots & \vdots\\ a_{n1} & a_{n2} & \dots & a_{nm}\\ \end{bmatrix}$$ [and so on].