Linear algebra is a branch of mathematics that is used by engineers and applied scientists to design and analyze complex systems. Civil engineers use linear algebra to design and analyze load-bearing structures such as bridges. Mechanical engineers use linear algebra to design and analyze suspension systems, and electrical engineers use it to design and analyze electrical circuits. Electrical, biomedical, and aerospace engineers use linear algebra to enhance X rays, tomographs, and images from space. In this chapter and the next we study two common problems from electrical engineering and use linear algebra to solve them. The two problems are (i) electrical circuit analysis and (ii) coordinate transformations for computer graphics. The first of these applications requires us to understand the solution of linear systems of equations, and the second requires us to understand the representation of mathematical operators with matrices.
Much of linear algebra is concerned with systematic techniques for organizing and solving simultaneous linear equations by elimination and substitution. The following example illustrates the basic ideas that we intend to develop.
Example 1
A woman steps onto a moving sidewalk at a large airport and stands while the moving sidewalk moves her forward at 1.2 meters/seconds. At the same time, a man begins walking against the motion of the sidewalk from the opposite end at 1.5 meters/second (relative to the sidewalk). If the moving sidewalk is 85 meters long, how far does each person travel (relative to the ground) before they pass each other?
To solve this problem, we first assign a variable to each unknown quantity. Let
Our second equation is based on the time required before they pass. Time equals distance divided by rate, and the time is the same for both people:
We may substitute Equation 2 into Equation 1 to obtain the result
Combining the result from Equation 3 with that of Equation 1, we find that
So the man travels 17 meters, and the woman travels 68 meters.
Equation 1 and Equation 2 are the key equations of Equation 1. They may be organized into the “matrix equation”
The rules for matrix-vector multiplication are evidently
Equation 2 and Equation 3 may be organized into the matrix equation
This equation represents one partially solved form of Equation 5, wherein we have used the so-called Gauss elimination procedure to introduce a zero into the matrix equation in order to isolate one variable. The MATLAB software contains built-in procedures to implement Gauss elimination on much larger matrices. Thus MATLAB may be used to solve large systems of linear equations.
Before we can apply linear algebra to more interesting physical problems, we need to introduce the mathematical tools we will use.
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