Math 215 - Fall 2002

Homework Assignments

Homework #9 Due Weds Nov 27 11:30 (in class)

6.5 #12 (feel free to use Mathematica!), 19, 20

6.6 #4 (again, use Mathematica to make this easy -- you don't have to graph your solution)

6.7 #3,5,7, 13, 17, 22, 25

Homework #8 Due Thursday, Nov 21 Noon

6.2 #6, 10, 13, 27, 29 (remember an orthogonal matrix has orthoNORMAL columns)

6.3 #4, 9, 13, 23, 24

6.4 #9

Chapter 6 Supplementary (p. 444) #4, 6, 8
Chapter 5 Supplementary (p. 370) #2, 3, 6, 7



Special Problem #7, Due Thursday, Dec 12 Noon (LAST ONE!!)

Read Section 5.6 (Discrete Dynamical Systems)  This gives some realistic examples of diagonalization and complex eigenvalues.

Then do Problems #2, 5, 6, 8 in section 5.6.

Homework #7 Due Thursday, November 14, NOON

Eigenthings Worksheet #2
5.1 #23, 26
5.2 #2, 9
5.3 #8, 10, 12 (has evalues 2 and 8), 17, 24, 27
6.1 #2, 13, 16, 24, 25, 30

Note: You may use Mathematica to check your work:

Eigenvalues[A] gives the eigenvalues of A.
Eigenvectors[A] gives the eigenvectors of A.
Eigensystem[A] gives the both eigenvectors and eigenvalues of A.
To take the inner product of two vectors, use ".", as in {1,2}.{3,4}, which evaluates to 1


Special Problem #6 (Due Thursday, Dec 5, Noon):

Section 4.3 #38 (Set up the dependence relation, choose several specific values of t to get 7 equations and 7 unknowns and have Mathematica solve them.)

Section 4.5 #34 (Again, Mathematica will be a big help here)

Section 4.7, #17, 18 (The idea here is that integrals involving powers of cosine are hard, but those involving cos(at) are easy.  By doing a change of basis you can express the hard functions as linear combinations of the easy ones.  Be sure to use Mathematica to do the hard work, such as finding the inverse of P.)
Homework #6 Due Thursday, October 31, NOON

4.4 #1, 5, 13, 19

4.2 #35 (this is needed for 4.5 #31 below)

4.5 #1, 8, 10, 14, 31 (hint:  show that if {b1, ..., bn} is a basis for H then {T(b1), ..., T(bn)} spans T(H), and use the spanning set theorem), #32 (hint: same as in 31, except now show that one-to-oneness of T means that {T(b1), ..., T(bn)}  is also linearly independent -- a proof of something similar was required on the last take-home exam)

4.6 #5, 8, 9, 19
Special Problem #5 (Due Thursday, Nov 21,  Noon):

1. Read Section 4.9 on Markov Chains.  This is one of the most useful applications of matrices in the real-world.

2. Do exercise 8 in section 4.9.

3. Do exercise 17 in section 4.9 (the steps of the proof are given.  Your job is to explain each one, quoting an appropriate theorem, if applicable.)




Homework #5 Due Thursday, October 24, NOON

4.1 #32, 33

4.2 #6, 7, 13,  24, 31, 33 (this problem continues on p. 236)

4.3 #14, 29, 30, 34

There is NO special problem for this week.


Homework #4  Due Thursday, October 10 NOON

2.3 #15, 18, 20, 26 (A^2 is A times A), 27, 38

Chap 2 Supplementary (p. 183) #2, 3, 4 (use #3 as a guide), 10, 18

3.1 #2 (do it twice: expand across first row and down second column), 10, 41 (the area of a parallelogram is its base times its height)

3.2 #16, 18, 20, 26, 33, 34, 36

4.1 #3, 5, 7, 8, 12, 16, 21, 28

Special Problem #4 (Due Thursday, October 31 (boo!) Noon):

Let A be the matrix K+J, where K is the nxn identity matrix and J is the nxn matrix entirely of 1's.  For example, when n=4, A looks like

2 1 1 1
1 2 1 1
1 1 2 1
1 1 1 2.

a) Compute det(A) when n=4.

b) Find a formula for det(A) that works for arbitrary n.

c) Let C = aK+bJ, where K and J are as above, and a and b are arbitrary scalars.  Find a formula for det(C).  Your answer should depend on a, b and n.  You don't have to prove that the formula works, but you should explain how you found it.

Hint:  try to find some simple row operations that introduce the largest possible number of zeroes.  You might also want to compute some examples with Mathematica.  To easily construct an nxn identity matrix you can use

K = IndentityMatrix[n]

where n is an integer.  To construct the matrix J you can write

J =  Table[1, {i, n}, {j, n}]

where you'll want to replace n by an integer.

Homework #3 (Due Noon Thurs, Sept 26)

1.9 #5, 10, 18, 25, 27, 30, 33, 36
 

2.1 #1, 8, 9, 10, 21, 22, 33
 

2.2 #2, 15, 20, 31, 33, 34


Special Problem #3 (Due Tuesday, October 22, 12 noon):  The "trace" of a square matrix is the sum of the entries along the main diagonal (i.e. from top left to bottom right).
 

a)  Make up an example of a 3x2 matrix A and a 2x3 matrix B and show that the trace of AB = trace of BA

b) Prove that if A and B are matrices which can be multiplied in both orders then, the trace of AB = trace BA.
 


Homework #2 (Due 9pm Weds, Sept 18):

Special Problem #2: (Due October 10, 12 noon.)  Special problems should be handed in separately from your regular homework, in a separate tray outside Hilles 207. Remember these can be handed in multiple times, for hints and corrections, with the only the final submission counting for credit.

A Vandermonde matrix is a square matrix whose entries in each row are successive powers of a scalar, starting with the power 0.
So a 2x2 Vandermonde matrix looks like
 
1 a
1 b

and a 3x3 Vandermonde matrix looks like
 
 
1 a a^2
1 b b^2
1 c c^2

(where a^2 is "a squared", etc).

a) Show that the columns of the above 2x2 Vandermonde matrix are linearly independent if and only if a is not equal to b.

b) Show that the columns of the above 3x3 Vandermonde matrix are linearly independent if and only if a, b, and c are all distinct.

c) Formulate and prove the corresponding statement about 4x4 Vandermonde matrices.
 


Homework #1 (Due 9pm Weds, Sept 11):
 


Special Problem #1: (Due October 3, 12 noon.)  Special problems should be handed in separately from your regular homework, in a separate tray outside Hilles 207.

a) Learn how to use Mathematica to enter matrices, row reduce them and solve linear systems.  You can download a tutorial notebook from

http://www.haverford.edu/math/jtecosky/215/intromma.html
 
 

b) Section 1.2 #34.  Use Mathematica to do the gruntwork once you figure out what to do. (Read the explanation preceding problem #33).

After you have found the coefficents a0, a1, a2, a3, a4, a5 you should have Mathematica graph the polynomial and the six points on the same graph to see if the polynomial passes through all of the points.  Here is the command which plots a quintic and the given points.  The quintic polynomial given is just an example to show you where to fill in your polynomial -- it doesn't pass through any of the points, but if you replace it with one using the coefficients you found, the graph should pass nicely through all six points.
 
 

Plot[7 + 6t - t^2 - t^3 + t^4 - 2t^5, {t, 0, 10},
    PlotRange -> {0, 120},
    Epilog -> {PointSize[0.02], Point[{0, 0}], Point[{2, 2.9}],
        Point[{4, 14.8}], Point[{6, 39.6}], Point[{8, 74.3}],
        Point[{10, 119}] }]

This command is included in the tutorial notebook listed in part a), so you can cut and paste it into your own notebook without having to type it from scratch.

Hand In:  A printout of your mathematica session (with your name, of course) showing your work and the graph of the interpolating polynomial passing through all 6 points.
 


Homework #0 (Due Weds, Sept 4, not graded) :