Math 322

These problems are provided to help you study. The presence of a
problem on this handout does not imply that there *will* be a
similar problem on the test. And the absence of a topic does not
imply that it *won't* appear on the test.

1. Consider the real matrix

Find the eigenvalues and a complete set of independent eigenvectors, and a matrix P such that is diagonal, and the corresponding diagonal matrix.

2. Consider the real matrix

Find the eigenvalues and a complete set of independent eigenvectors, and a matrix P such that is diagonal, and the corresponding diagonal matrix.

3. Give an example of a nonzero matrix over which is not diagonalizable.

4. Suppose that is not a multiple of . Prove that the real matrix A which gives rotation counterclockwise through does not have any real eigenvalues.

(You can do this algebraically, but see if you can give a geometric argument.)

5. Suppose and *every* vector in
is an eigenvector of A. Prove that A is a multiple of the identity matrix.

6. Let A be an matrix, let v be an eigenvector corresponding to the eigenvalue , and let . Prove or disprove: is an eigenvector of A.

7. Let , and let

Show that is an eigenvector for M corresponding to the eigenvalue .

8. Find the general solution to each of the following differential equations.

(a) .

(b) .

(c) .

9. Solve the following linear system for x, y, and z in terms of t:

10. Solve the following linear system for x and y in terms of t:

Your answer should be given entirely in terms of real numbers and functions.

11. Two tanks hold 100 gallons of liquid each. The first tank starts with 36 pounds of dissolved salt, while the second starts with pure water. Pure water flows into the first tank at 5 gallons per minute; the well-stirred mixture flows into tank 2 at 9 gallons per minute. The mixture in tank 2 is pumped back into tank 1 at 4 gallons per minute, and also drains out at 5 gallons per minute. Find the amount of salt in each tank after t minutes.

12. Suppose that A is a real matrix and

Find A.

13. Find for

14. Compute for

15. Compute for

16. Compute for

17. Let denote the standard inner product on .

(a) Find .

(b) Find the cosine of the angle between and .

(c) Find a nonzero vector that is perpendicular to both and .

18. Suppose that u, v, and w are vectors in a real inner product space, and

(a) Find .

(b) Find .

19. Find an orthonormal basis relative to the standard dot product on for the subspace spanned by the set

20. Let denote the real vector space of continuous real-valued functions on the interval . An inner product is defined on by

(a) Compute , where and .

(b) Find , where .

(c) For what value of k are the functions and orthogonal?

(d) Consider the set of functions . Find an orthonormal set which spans the same subspace of as S.

21. The following set of vectors in is orthonormal relative to the standard dot product:

Find the components of relative to the basis .

22. An inner product is defined on by

(a) Find relative to the given inner product.

(b) Find , where is the angle relative to the given inner product between and .

(c) Find a nonzero vector which is orthogonal to relative to this inner product.

23. Let x be a fixed vector in a real inner product space V. Let

Prove that is a subspace of V. ( is called the
* orthogonal complement* of x.)

1. Consider the real matrix

Find the eigenvalues and a complete set of independent eigenvectors, and a matrix P such that is diagonal, and the corresponding diagonal matrix.

The characteristic polynomial is

The eigenvalues are and .

For ,

With a, b, and c as variables, the corresponding homogeneous system is , or . The solution vector is

Taking and , and then and , I get the eigenvectors and .

For ,

With a, b, and c as variables, the corresponding homogeneous system is , or , and . The solution vector is

Taking , I get the eigenvector .

Using the eigenvectors as columns, I obtain

2. Consider the real matrix

The eigenvalues are 2 and .

For ,

With eigenvector , the row reduced echelon matrix gives the equations

So

Hence, is an eigenvector for .

For ,

The last two rows are clearly independent of the first, so they must be multiples (or all three rows would be independent, and there would be no nonzero eigenvectors). It follows that some row operation can be used to "wipe out" the third row (in fact, works), and

(Note that you don't need to actually figure out the row operation
that does this --- you *know* that one exists, because the
rows *must* be multiples.)

With eigenvector , the row reduced echelon matrix gives the equations

The first equation gives . The second equation is satisfied by and (by swapping the coefficients " " and "5" and negating one of them, in this case the " "). Thus, is an eigenvector for .

By conjugation, is an eigenvector for .

(At this point, note that you might get different looking, but correct, results if you had "wiped out" the second row of the matrix above rather than the third row, or if if you had chosen to negative the "5" rather than the " ".)

A diagonalizing matrix is

3. Give an example of a nonzero matrix over which is not diagonalizable.

Consider the matrix

The characteristic polynomial is , so is the only eigenvalue. Now

With as a solution vector, the corresponding homogeneous system is . Thus,

Taking , I get the eigenvector .

Since there is only one eigenvector, the matrix is not diagonalizable.

4. Suppose that is not a multiple of . Prove that the real matrix A which gives rotation counterclockwise through does not have any real eigenvalues.

First, rotation through is an invertible operation (the inverse is rotation through ). Hence, A is invertible, and 0 can't be an eigenvalue of A.

Next, if c is a nonzero eigenvalue of A with eigenvector v, then . This equation says that and are parallel; since and v are parallel, this means that and v are parallel. But this is impossible, because is just v rotated by , which is not a multiple of .

Hence, A has no nonzero eigenvalues.

5. Suppose and *every* vector in
is an eigenvector of A. Prove that A is a multiple of the identity matrix.

I need a little fact about matrix multiplication which you should check for yourself: If is the standard basis vector, then is the column of A. (Try it out for a matrix to see the idea.)

First, I'll show that A is a diagonal matrix. Every vector is an eigenvector, so in particular the standard basis vectors are eigenvectors. So

But is the first column of A, so this says that the first column of A is

Likewise,

But is the second column of A, so this says that the second column of A is

Continue with , ..., . Then stringing the columns of A together, I find that

But the vector is also an eigenvector of A, so for some ,

Multiplying out the two sides of this equation, I get

So , , ... are all equal to , and

This is just times the identity matrix.

6. Let A be an matrix, let v be an eigenvector corresponding to the eigenvalue , and let . Prove or disprove: is an eigenvector of A.

Since v is an eigenvector corresponding to the eigenvalue ,

Multiply both sides by c:

Since and (since v is an eigenvector), it follows that . Hence, is an eigenvector for A corresponding to the eigenvalue .

7. Let , and let

Show that is an eigenvector for M corresponding to the eigenvalue .

The definition says that x is an eigenvector for a matrix A with
eigenvalue if , and that is what I've shown for
M, , and . I *don't* need to go through a lot
of trouble computing the characteristic polynomial of M and finding
eigenvalues and eigenvectors. Did you try to do that?

8. Find the general solution to each of the following differential equations.

(a) .

(b) .

(c) .

(a) In operator form, this is , or . The solution is

(b) The root 2 is repeated 3 times; the "D" term corresponds to a root of 0. The general solution is

(c) In operator form, this is . The equation has roots . The general solution is

9. Solve the following linear system for x, y, and z in terms of t:

Let

The characteristic polynomial is

The eigenvalues are and .

For :

With variables a, b, and c, the homogeneous system is , or . So

This gives the independent eigenvectors and .

For :

With variables a, b, and c, the homogeneous system is

Then and . So

Taking , I get the eigenvector .

The general solution is

10. Solve the following linear system for x and y in terms of t:

Your answer should be given entirely in terms of real numbers and functions.

Let

The characteristic polynomial is

The roots are .

For , I have

Since the rows must be multiples, I can drop the second row. With variables a and b, the first row yields the homogeneous system

By inspection, an eigenvector is .

The solution corresponding to this eigenvector is

The real and imaginary parts give two independent solutions:

They give the general solution

11. Two tanks hold 100 gallons of liquid each. The first tank starts with 36 pounds of dissolved salt, while the second starts with pure water. Pure water flows into the first tank at 5 gallons per minute; the well-stirred mixture flows into tank 2 at 9 gallons per minute. The mixture in tank 2 is pumped back into tank 1 at 4 gallons per minute, and also drains out at 5 gallons per minute. Find the amount of salt in each tank after t minutes.

Let x be the number of pounds of salt dissolved in the first tank at time t and let y be the number of pounds of salt dissolved in the second tank at time t. The rate equations are

Simplify:

Next, find the characteristic polynomial:

The eigenvalues are , .

Consider :

is an eigenvector.

Now consider :

is an eigenvector.

The solution is

When , and . Plug in:

Solving for the constants, I obtain , . Thus,

12. Suppose that A is a real matrix and

Find A.

Setting , I have

13. Find for

Here's a summary of the algorithm for computing .

Let be the eigenvalues of A.

(If the characteristic polynomial has a multiple root, you list the eigenvalue multiple times. For instance, if the characteristic polynomial has a factor of , you list the eigenvalue 5 three times.)

Define

Then

Note that in defining the , you never get to the last eigenvalue . But *does* occur in .

If you have complex roots, the result may have complex numbers in it until you do some simplification.

14. Compute for

The characteristic polynomial is

The eigenvalue is (double), so my list of eigenvalues is .

First,

Next, and

Thus,

15. Compute for

Since the matrix is upper triangular, the eigenvalues are the diagonal elements 3 and -2.

I have

Next, and

So

16. Compute for

Since the matrix is lower triangular, the eigenvalues are the diagonal entries 1 (double) and 2. I will list the eigenvalues in this order: .

I have

Moreover,

Here's the work for the third integral, which is done by parts:

Hence,

17. Let denote the standard inner product on .

(a) Find .

(b) Find the cosine of the angle between and .

(c) Find a nonzero vector that is perpendicular to both and .

(a)

(b)

(c) I want such that

This gives the equations

Row reduce to solve:

The parametric solutions are

I can get a nonzero solution by setting at least one of s, t equal to a nonzero value. For example, taking and , I get

18. Suppose that u, v, and w are vectors in a real inner product space, and

(a) Find .

(b) Find .

(a)

(b) Note that . Now

Hence, .

19. Find an orthonormal basis relative to the standard dot product on for the subspace spanned by the set

First, I'll get an orthogonal basis for the subspace. At the end, I'll divide each vector by its length to get an orthonormal set. To get the orthogonal basis, apply Gram-Schmidt to the original set of vectors.

The first vector in the orthogonal set will be . The second vector is

The third vector is

To get the orthonormal basis, divide , , and by their lengths. The orthonormal basis is

20. Let denote the real vector space of continuous real-valued functions on the interval . An inner product is defined on by

(a) Compute , where and .

(b) Find , where .

(c) For what value of k are the functions and orthogonal?

(d) Consider the set of functions . Find an orthonormal set which spans the same subspace of as S.

(a)

(b)

Hence, .

(c)

Setting , I have

(d) I have to do Gram-Schmidt on the set S. The first vector will be .

For the second vector, I have

I need to compute the integrals in the top and bottom of the fraction:

Returning to , I have

I may multiply by 7 to clear denominators to obtain .

Finally, I have to divide and by their lengths. Since , I know . Also,

So . The orthonormal set is

21. The following set of vectors in is orthonormal relative to the standard dot product:

Find the components of relative to the basis .

Since the basis is orthonormal, I can find the components by taking the dot product of with each of the vectors:

That is,

22. An inner product is defined on by

(a) Find relative to the given inner product.

(b) Find , where is the angle relative to the given inner product between and .

(c) Find a nonzero vector which is orthogonal to relative to this inner product.

(a)

(b)

Now

Hence,

(c) Find a nonzero vector which is orthogonal to relative to this inner product.

I want

Multiplying out gives the equation

A nonzero solution is given by .

23. Let x be a fixed vector in a real inner product space V. Let

Prove that is a subspace of V. ( is called the
* orthogonal complement* of x.)

I must show that is closed under sums and under scalar multiplication.

Let . Then

Therefore, .

Let and let . Then

Therefore, .

Hence, is a subspace of V.

*If we can really understand the problem, the answer will come out
of it, because the answer is not separate from the problem.* -
*Jiddu Krishnamurti*

Copyright 2020 by Bruce Ikenaga