I have been given the following task (by my professor, with no mentionable context):
Prove that $\displaystyle \left[ \begin{array}{rr} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array} \right] $ $\displaystyle \left[ \begin{array}{rr} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{array} \right] = \left[ b_{11}A_1+b_{21}A_2 : b_{12}A_1+b_{22}A_2 \right]$, where $\displaystyle A_1 = \left[ \begin{array}{r} a_{11} \\ a_{21} \\ \end{array} \right]$ and $\displaystyle A_2= \left[ \begin{array}{r} a_{12} \\ a_{22} \\ \end{array} \right]$. Generalise the result to $n \times n$ matrices.
However, I don't know what is implied by the notation:
$\left[ b_{11}A_1+b_{21}A_2 : b_{12}A_1+b_{22}A_2 \right]$
So, I can't reasonably hope to satisfy the desired results, when I am uncertain what exactly is desired.
What is implied by this?
It certainly means that what's on the left of ":" is the first column of the matrix that results from the product, and what's on the right is the second column.