The Dot Product for Vectors in Rn
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
Find the dot product of
a = [2 1 0 6 -1] and b =
a . b = (2)(5) + (1)(2) + (0)(-3) + (6)(0) + (-1)(1) = 11
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
Then the matrix product is
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
can be written as
Often, we write the matrix equation in augmented form as shown below