MATH 280, Spring 2002 MATH 280, Spring 2002

Theme I: Solutions of Linear Equations



The due date for this theme is Friday, January 25, at noon.

Reading assignment: Section 1.4, 1.5

Background reading: Section 1.3



We have discussed in class equations involving vectors

x1a1+x2a2+¼xnan = b
in which a1,¼,an and b are vectors and the x1,¼,xn are scalars and have seen that this system has a solution if b can be written as a linear combination of a1,¼,an. In this theme, we will investigate matrix equations of the form
Ax = b
in which A is a matrix and b and x are vectors.

To gain some familiarity with the notation, practice multiplying matrices and vectors using the following problems. In each case, first decide whether it is possible to perform the indicated multiplication.

é
ê
ë
2
5
-1
3
 ù
ú
û 		é
ê
ë
4
-1
 ù
ú
û
é
ê
ë
3
7
0
2
-5
4
 ù
ú
û 		é
ê
ê
ê
ê
ë
2
4
-3
 ù
ú
ú
ú
ú
û
é
ê
ê
ê
ê
ë
3
1
-2
3
-4
5
 ù
ú
ú
ú
ú
û 		é
ê
ê
ê
ê
ë
5
2
2
 ù
ú
ú
ú
ú
û
é
ê
ê
ê
ê
ê
ë
6
4
5
3
4
2
3
1
 ù
ú
ú
ú
ú
ú
û 		é
ê
ë
5
2
 ù
ú
û

Which multiplications could not be performed? Why?

Our goal is to understand the solution space of the equation Ax = b. To begin, explain why the matrix equation Ax = b has a solution if and only if b is a linear combination of the columns of A. In discussing your answer, you will need to explain a linear combination of vectors and the span of a set of vectors. Give a numerical example of a matrix A and a vector b such that Ax = b is not consistent, and give explicit details to show that the equation cannot be solved. To gain practice for answering these questions, work problems 11, 17, 19, 31 on page 46.

When is a matrix equation homogeneous? Why does the work on the previous paragraph say that a homogeneous equation always has at least one solution? What is it? When does a homogeneous equation have more than one solution? Give a numerical example (i.e., a matrix A and a vector b) demonstrating this condition. Suppose the homogeneous equation has a nontrivial solutions u. What property of matrices or vectors guarantees that cu is also a solution, for any scalar c?

Let

A = é
ê
ê
ê
ê
ë
1
2
-1
1
0
3
3
2
5
 ù
ú
ú
ú
ú
û
and
b = é
ê
ê
ê
ê
ë
4
-1
2
 ù
ú
ú
ú
ú
û 		.
Consider the matrix equation Ax = b and the corresponding homogeneous equation Ax = 0. Use these equations to address the following problems. Does Ax = b have a solution? Find one. Find the solution set of the corresponding homogeneous equation Ax = 0. Let v1 and v2 be (arbitrary) solutions of the homogeneous equation and let p1 and p2 be (arbitrary) solutions of Ax = b. Consider the three vectors v1+v2, v1+p1 and p1+p2. Which of these vectors are solutions of the original matrix equation, homogeneous equation, or neither? What property of matrices or vectors guarantees your answers. Why? Use specific vectors you found above to demonstrate your answers.

Use matrix methods to solve the following problem:

Three species of bacteria will be kept in one test tube and will feed on three resources. Each member of the first species consumes 3 units of the first resource and 1 unit of the third per day. Each bacterium of the second type consumes 1 unit of the first resource and 2 units each of the second and third. Each bacterium of the third type consumes 2 units of the first resource and 4 each of the second and third. If the test tube is supplied daily with 12,000 units of the first resource, 12,000 units of the second and 14,000 units of the third, how many of each species can coexist in equilibrium in the test tube so that all of the supplied resources are consumed? Explain all of your work.

File translated from TEX by TTH, version 1.95.
On 18 Jan 2002, 12:49.