[skedaddle]

In the row picture, the elements of a matrix are the coefficients of linear functions acting on its domain. In the column picture they are coordinates in its image.

Any matrix A or B can be interpreted from either point of view on its own. When you take their product AB, each of A's functions (row picture) is evaluated on each of B's points (column picture).

This gives an associative (but not commutative) algebra. If you go around the column picture with an operator like A.B=AB^T, you get

    (A.B).C = (AB^T).C
            = AB^TC^T
    

    A.(B.C) = A.(BC^T)
            = A(BC^T)^T
            = ACB^T

The two formulas are not equal, and second involves "traditional" matrix multiplication. You can compute products like this operationally though, as long as you work from left to right.