Left Multiplication Transforms Rows, Right Multiplication Transforms Columns

Why left multiplication is a row transformation and right multiplication is a column transformation

For the matrix product AB=C, the rows of C are linear combinations of the rows of B
Likewise, the columns of C are linear combinations of the columns of A, so right-multiplying by B amounts to applying a column transformation to A.

AB=C

• A can be seen as the coefficient matrix, and B as the row matrix

• The first row of A holds the coefficients for the first row of C
  as shown below:
  

  Here A is the coefficient matrix, i.e. the coefficients of the row transformation applied to B
  

• Treat B as the coefficient matrix and A as the column matrix (each column of C is a linear combination of the columns of A)

• The first column of B holds the coefficients for the first column of C
  as shown below:
  
  

Overall figure: