Matrix Multiplication and Coordinate Systems
1. Matrix Multiplication
1.1 Definition
A matrix multiplication is the production of a new matrix from a pair of matrices.
Matrices can only multiply if the number of columns for the first matrix equals the number of rows for the second matrix.
For example
Multiplying AB
A ---> 3x2 matrix (3 is the # of rows, and 2 is the # of columns)
B ---> 2x3 matrix (2 is the # of rows, and 3 is the # of columns)
THEY DO CAN MULTIPLY!
The new matrix will have the rows of the first matrix and the columns of the second matrix.
For example
AB = C
A ---> "m x p"
B ---> "p x n"
Then C will be "m x n"
1.2 Dot Product
$ A= \left(\begin{array}{cccc}a1\\a2\\.\\.\\.\\an\end{array}\right) $ $ B= \left(\begin{array}{cccc}b1\\b2\\.\\.\\.\\bn\end{array}\right) $
A*B = a1b1 + a2b2 + ... + anbn
Matrix multiplication is a way to combine two matrices and get a third matrix. To find this third matrix, you have to compute each entry in the third matrix one at a time. So to find the entry (a,b) in the third matrix, you take the sum of the products of the elements in the ath row in the first matrix and the bth column in second matrix. Suppose the ath row equals [a1,a2,...,an] and the bth column equals [b1,b2,...,bn]. Then we can compute every entry (a,b) of the third matrix as (a,b) = a1*b1 + a2*b2 + ... + an*bn.
