Careful Definitions

You may be uneasy with mathematical operations being defined in this casual fashion. Here are some more accurate definitions:

Matrix times Column Matrix: Let A be a M × C matrix, and x be a C × 1 column matrix. Then the product Ax is the M × 1 column matrix whose i-th element is the dot product of row i of A with x.

Row Matrix times Matrix: Let x be a 1 × R row matrix and A be a R × M matrix. Then the product xA is the 1 × M row matrix whose i-th element is the dot product of x with column i of A.