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We are now equipped to set up systems of linear equations as matrix- vector equations so that they can be solved in a standard way on a computer.Suppose we want to solve the equations from Example 1 from "Linear Algebra: Introduction" for and using a computer. Recall that Equations 1 and 2 from Linear Algebra: Introduction are
The first step is to arrange each equation with all references to in the first column, all references to in the second column, etc., and all constants on the right-hand side:
Then the equations can be converted to a single matrix-vector equation. The coefficients form a matrix, keeping their same relative positions, and the variables and constants each form a vector:
Equation 3 is of the general form
where in this case
Given any and , MATLAB can solve Equation 4 for (as long as a solution exists). Key ideas in the solution process are the identity matrix and the inverse of a matrix.
When the matrix A is the matrix , the vector is the l-vector , and the vector is the l-vector , then the matrix equation becomes the scalar equation
The scalar is the inverse of the scalar , so we may multiply on both sides of Equation 6 to obtain the result
We would like to generalize this simple idea to the matrix equation so that we can find an inverse of the matrix , called , and write
In this equation the matrix is the identity matrix
It is clear that the identity matrix and the inverse of a matrix, , play a key role in the solution of linear equations. Let's study them in more detail.
The Matrix Identity. When we multiply a scalar by 1, we get back that same scalar. For this reason, the number 1 is called the multiplicative identity element. This may seem trivial, but the generalization to matrices is more interesting. The question is, is there a matrix that, when it multipliesanother matrix, does not change the other matrix? The answer is yes. The matrix is called the identity matrix and is denoted by . The identity matrix is always square, with whatever size is appropriate for the matrix multiplication.The identity matrix has l's on its main diagonal and everywhere else. For example,
The subscript 5 indicates the size. In terms of the unit coordinate vectors , we can also write the identity matrix as
For any matrix , we have
For , we obtain the following special form for any vector :
This last equation can be written as the sum
This is a special case of one of the results you proved in Exercise 3 from "Linear Algebra: Matrices" . Figure 1 illustrates Equation 14 and shows how the vector can be broken into component vectors , each of which lies in the direction of one coordinate axis. The values of the are the coordinates of , and their magnitudes are also the lengths of the component vectors.
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