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
In order to make a matrix multiplication, there should be the operation of dot product between the rows of the first matrix and the columns of the second matrix. To fin the entry (a,b) in the new matrix, the sum of the products of the bth column in the second matrix and of the ath row in the first matrix.
For example
<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math>
<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math>
You take a11*b11 + a12*b21 and that's your c11.
(1)*(5) + (2)*(7) = 19
Then a11*b12 + a12*b22 = c12
(1)*(6) + (2)*(8) = 22
After no more columns, you move to the next row. Take a21*b11 + a22*b21 = c21
(3)*(5) + (4)*(7) = 43
And the last one in this case would be a21*b12 + a22*b22 = c22
(3)*(6) + (4)*(8) = 50
So, the new matrix would be
<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math>
1.3 Matrix Multiplication in the Real World
Matrix multiplication are used in many life situations. In fact this semestre I had two clases where matrices where applied.
To solve for variables and to get probabilities are some of the main functions in the real world.
When you have equations, and need to solve for the unknown variables, a matrix multiplication and rref let you do this.
For example
Equation 1 -------> 12x + 3y = 42
Equation 2 -------> 3x + 8y = 54
Equation 3 -------> 15x + y = 36
You can put them in a matrix multiplication like this:
<math>\left(\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right)</math>
You can verify that the equations and the matrix are equal by doing the dot product.
After doing the rref, you get that the variable x= 2 and y = 6 .
1.4 Matrix Multiplication Related to Other Classes.
