Matrix Multiplication
The Dot Product for Vectors in R^{n} Let a = [a_{1} a_{2} ... a_{n}] be a vector in R^{n} (considered as a 1 by n matrix) and let then the dot product of a and b is defined by a ^{.} b = a_{1}b_{1} + a_{2}b_{2} + ... + a_{n}b_{n} = S a_{i}b_{i}
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} = v_{i}_{ }._{ }w_{j} 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, x^{T} = (x_{1} x_{2} ... x_{n}) 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
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