Matrix Multiplication as Linear Combination
Matrix-Vector Multiplication
Suppose we have a matrix
A matrix right-multiplied by a column vector gives a column vector, which is the linear combination of its columns.
Suppose we have a row vector
A matrix left-multiplied by a row vector gives a row vector, which is the linear combination of its rows.
Matrix-Matrix Multiplication
Suppose we have a matrix
An
An
In one word,
Rank of Matrix Product
The rank of matrix
similarly we have
Therefore