Matrix Multiplication   The Dot Product for Vectors in Rn Let         a  =  [a1 a2 ... an]  be a vector in Rn (considered as a 1 by n matrix) and let     then the dot product of a and b is defined by         a .  b  =  a1b1 + a2b2 + ... + anbn  = S aibi   Example Find the dot product of        a  =  [2  1  0  6  -1]   and    b  =   Solution We have         a . b  =  (2)(5) + (1)(2) + (0)(-3) + (6)(0) + (-1)(1)  =  11   Matrix Multiplication There are many ways of thinking about a matrix.  One way is as a collection of row vectors and another way is as a collection of column vectors.  Consider the m by p matrix A (considered as a matrix of row vectors) and the p by n matrix B (considered as a matrix of column vectors).  The matrices are shown below.         We define the matrix product by         (AB)ij  =  vi . wj Remark: If the number of columns of A is not equal to the number of rows of B, then the product AB is not defined. Remark: It is not true in general that AB and BA are the same matrix even if they are both defined. We can also define         Example Let         Then the matrix product is         Linear Systems Any m by n linear system can be written in the form          Ax  =  b Where A is the coefficient matrix,          xT  =  (x1  x2  ...  xn) and b is the m by 1 matrix of numbers to the left of the equality.  For example the linear system         2x + 3y + z  =  0         3x - 4y - z  =  6         x  + 2y + 3x  =  2 can be written as          Often, we write the matrix equation in augmented form as shown below         Back to the Linear Algebra Home Page