5.7 Linear algebra: solving linear systems of equations

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 $x_1$ and $x_2$ 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 $x_1$ in the first column, all references to $x_2$ 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 $A\in R^{n\times n}$ and $b\in R^n$ , MATLAB can solve Equation 4 for $x$ (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 $1\times 1$ matrix $a$ , the vector $x$ is the l-vector $x$ , and the vector $b$ is the l-vector $b$ , then the matrix equation $Ax=b$ becomes the scalar equation

The scalar $a^{-1}$ is the inverse of the scalar $a$ , 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 $Ax=b$ so that we can find an inverse of the matrix $A$ , called $A^{-1}$ , and write

In this equation the matrix $I$ is the identity matrix

It is clear that the identity matrix $I$ and the inverse of a matrix, $A^{-1}$ , 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 $I$ . 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 $0^{\text{'}}s$ everywhere else. For example,

The subscript 5 indicates the size. In terms of the unit coordinate vectors $e_i$ , we can also write the $n\times n$ identity matrix as

For any matrix $A\in R^{n\times p}$ , we have

For $p=1$ , we obtain the following special form for any vector $x\in R^n$ :

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 $x$ can be broken into component vectors $x_i{e}_i$ , each of which lies in the direction of one coordinate axis. The values of the $x_i^{\text{'}}s$ are the coordinates of $x$ , and their magnitudes are also the lengths of the component vectors.