Math 311-102 Assignments - Summer I, 2006
Assignment 1 - Wednesday, May 31.
- Read sections 1.1-1.4, 1.6 in the text.
- Do the following problems.
- Section 1.1: 9, 29
- Section 1.2: 30(a)
- Section 1.3: 18, 19
Assignment 2 - Due Thursday, June 1.
- Read sections 2.1A, 2.2 in the text. Also, read my Notes on Row Reduction.
- Do the following problems.
- Section 1.4: 21, 23
- Section 1.5: 27
- Section 1.6: 9, 16
- Extra problem. Let v = (1, -2, 1, 0) and
u = (2, 1, -1, 1).
- Find the lengths of v and u.
- Find the angle between v and u.
- Find the projection p of v onto u.
- Find the distance of the vector v to the line
x = tu.
Assignment 3 - Due Friday, June 2.
- Read section 2.2D, 2.3 in the text. Also, read these notes: The Rank of a Matrix.
- Do the following problems.
- Section 2.1A (pg. 51): 8
- Section 2.2C (pgs. 70-71): 17, 18, 21, 28
Assignment 4 - Due Monday, June 5.
- Read section 2.4 and 2.5 in the text.
- Do the following problems. The matrices you need are listed at
the end of this assignment.
- Section 2.2D (p. 73): 2, 11 (see errors)
- For the matrices R, S, u, v, find the combinations below, if
possible; if not, state why you can't.
(a) 6R+3S | (b)
RT | (c) RS | (d) SR | (e) vu
| (f) uv | (g) Ru |
- Find the reduced echelon form for the matrix B. In addition, find
rank(B) and find the leading columns of B.
- Use row reduction methods to solve the system S below. Put the
solution in parametric form. Identify the leading and non leading
variables. Find the solution to the corresponding homogeneous
system. Again, put the solution in parametric form.
-
Assignment 5 - Due Tuesday, June 6.
- Read section 2.5 and 3.1A in the text.
- Do the following problems. The matrices you need are listed at
the end of this assignment.
- A communications company does an analysis of discrete signals of
length five. They find that nearly all signals they encounter can be
represented as a linear combination of the three column vectors in the
set
S ={(1 0 2 −1 3)T, (1 −2 1 1 0)T,
(−1 1 0 2 −1)T}
This allows them to transmit three numbers rather
than five, except for the occasional signal that can't be represented
this way.
- Determine whether the discrete signal s = (4
−5 4 −1 4)T can be represented as a
linear combination of the three vectors. If so, find three numbers
that represent this signal.
- When a signal can be represented in this way, are the three
numbers unique? Explain. (Hint: are the vectors in S linearly
independent or dependent?)
- Section 2.3D (p. 81): 40.
- Section 2.4C (p. 87): 8, 23, 26.
- Either find the inverse of A below or show that it doesn't exist.
Assignment 6 - Due Wednesday, June 7.
- Read sections 3.1 and 3.2 in the text.
- Do the following problems.
- Section 2.5 (pgs. 98-99): 5, 6, 7, 11, 17, 18, 21
Assignment 7 - Due Thursday, June 8.
- Read sections 3.2 and 3.3 in the text.
- Do the following problems.
- Section 3.1 (pgs. 110-111): 5, 8, 12, 16, 19, 33
Assignment 8 - Due Friday, June 9.
- Read sections 3.2 and 3.3 in the text.
- Do the following problems.
- Section 3.2 (pgs. 118-119): 7, 8, 11, 13, 16, 22, 26
Assignment 9 - Due Tuesday, June 13.
- Read sections 3.3 and 3.4 in the text.
- Do the following problems.
- Section 3.3 (pgs. 125-126): 15, 16, 19, 20, 24, 25
Assignment 10 - Due Wednesday, June 14.
- Read section 3.5 in the text.
- Do the following problems.
- Section 3.4 (pgs. 130-131): 9, 10, 12, 13, 17, 20, 21(a,b)
Assignment 11 - Due Thursday, June 15.
- Read section 3.5 in the text and my notes,
Coordinate Vectors and Examples.
- Do the following problems.
- Let u1 = i + j -k,
u2 = 3i - j +k,
u3 = j + k. Show that B =
{u1, u2,
u3} is a basis for 3D space. Find [i +
2j]B.
- Determine whether {1, cos(2x), cos2(x)} is LD or LI.
- Let B = {ex, e−x}. Show that B is
LI. In addition, let V = span(B). Find [sinh(x)]B. (You may
need to look up the definition of sinh(x).)
- Show that C = {1 − 2x, 1 + 2x, 1 − x2} is a
basis for P2. Find [(2x −
1)2]C.
- Let L : P2 → P2 be defined by L[p] =
(x2 + 2)p'' + (x-1)p' - 4p.
- Show that L is linear.
- Find the matrix of L relative to the
basis B = {1, x, x2}.
- Find the matrix of L relative to the
basis C = {1 − 2x, 1 + 2x, 1 − x2}.
Assignment 12 - Due Friday, June 16.
- Read section 3.5 in the text and my notes,
Methods for Finding Bases.
- Do the following problems.
- Section 3.5B (pgs. 137-138): 10, 14, 17, 35
- Section 3.5C (pgs. 142-143): 4
- Let A be the matrix given below. Find the dimension of the image
of A. Use it and the Rank-Nullity Theorem (problem 11, §3.5C) to
find the dimension of the null space of A.
1 | -2 | 3 | 3 |
2 | -5 | 7 | 3 |
-1 | 3 | -4 | 3 |
- Suppose that B is a 7×10 matrix, and that the dimension of
the null space of B is 5. What is the dimension of the image of B
(i.e., column space)? What is the dimension of the image of
BT (row space of B)? What is the dimension of the null space of
BT?
Assignment 13 - Due Monday, June 19.
- Read section 3.6 in the text.
- Do the following problems.
- Section 3.6A (pg. 148): 6, 8, 17
- Let C be the matrix given below. Find bases for the column space,
null space, and row space of C, and state the rank and nullity of
C. What should these sum to? Do they?
1 | -3 | -1 | -3 |
-1 | 3 | 2 | 4 |
2 | -6 | 4 | 0 |
- Let L : P2 → P2 be defined by L[p]=
(2x2 + x + 1)p'' − (3x − 1)p' + 2p.
- Show that L is linear.
- Find the matrix A of L relative to the standard basis B = {1, x,
x2}.
- Find bases for the row space, null space, and image (column
space) of A.
- Use the bases you found above to write down bases for the null
space and image of L.
Assignment 14 - Due Tuesday, June 20.
- Read section 3.7 in the text.
- Do the following problems.
- Section 3.6A (pg. 148): 14
- Section 3.6C (pg. 154-155): 6, 9, 14, 16
Assignment 15 - Due Wednesday, June 21.
- Read section 3.7B in the text.
- Do the following problems.
- Let L : P2 → P2 be defined by L[p]=
p'' − 2xp'. You are given that L is linear. Solve the eigenvalue
problem for L. Does L have a basis relative to which its matrix is
diagonal?
-
Let f(x) = 1 -2 sin(x) and g(x) = 3 + cos(x). With the inner product
in Example 2, pg. 157, find the following quantities:
- ||f|| and ||g||
- < f, g >
- The angle between f and g.
- Section 3.7A (pg. 158): 2, 3, 13
Assignment 16 - Due Thursday, June 22.
- Read section 3.7B in the text.
- Do the following problems.
- In class we discussed the Gram matrix A for a least squares
problem. Given a linearly independent set of vectors
{w1,..., wn} and an inner product
< , >, we define the entries of A via
Aj,k = <wj, wk>.
For the vectors and inner products below, calculate the Gram matrix.
- {(1 1 1 1)T, (0 1 2 3)T,(0 1 4
9)T}, <x,y > =
yTx
- {(1 1 1 1)T, (-3 -1 1 3)T,(1 -1 -1
1)T}, <x,y > =
yTx
- {1,x,x2}, < f,g > =
∫01 f(x)g(x)dx
- {1, cos(x), sin(x)}, < f,g > =
∫-ππ f(x)g(x)dx
- Use the Gram matrix in (a) above to find the least squares
straight line fit to the data below
t = 0, 1, 2, 3; y = 1.1, 2.9, 5.0, 6.9
-
Use the Gram matrix and inner product in (c) to find the best
quadratic fit to y = e-x.
Assignment 17 - Due Friday, June 23.
- Read section 3.7B in the text.
- Do the following problems.
- For each of the following linearly independent sets of vectors
and inner products, use the Gram-Schmidt procedure to find an
orthogonal set with the same span as the original set.
- {(1 1 1 1)T, (0 1 2 3)T,(0 1 4
9)T}, <x,y > =
yTx
- {1,x,x2}, < f,g > =
∫01 f(x)g(x)dx
- {1, cos(x), sin(x)}, < f,g > =
∫0π f(x)g(x)dx (Note: the lower limit is
0, not π.)
- Use the inner product and orthogonal set from 1(b) above to find
the best quadratic fit to e2x. Leave your answer in terms of
the orthonormal polynomials you found in 1(b).
- Consider the inner product < f,g > =
∫-11 f(x)g(x)dx. In class we found the first
three normalized Legendre polynomials,
p0(x) = 2-½, p1(x) =
(3/2)½x, and p2(x) =
(5/8)½(3x2 − 1).
These form an orthonormal basis for P2 with respect
to the inner product. Use the Gram-Schmidt procedure to find
p3(x), the degree 3 normalized Legendre polynomial. This
gives an orthonormal basis for P3.
Assignment 18 - Due Tuesday, June 27.
- Read section 14.8 in the text.
- Do the following problems.
- Section 14.8 (pg. 713): 5, 6
Assignment 19 - Due Wednesday, June 28.
- Read section 14.7 in the text and the Notes
on Special Functions.
- Do the following problems.
- Section 14.8 (pg. 713): 3, 8, 13, 18.
Assignment 20 - Due Thursday, June 29.
- Read section 9.6 and the Notes
on Special Functions.
- Do the following problems.
- Use the method of Frobenius in the following differential
equations to find the indicial equation, the recurrence (recursion) relation,
and the the first few terms of the series solution for the largest
root of the indicial equation. (You are not being asked to
solve the recurrence relation.)
- x2y''+xy' +(x2 − 1)y = 0
- 9x2y'' +(9x2 +2)y = 0
- 25x2y''+25xy'+(x4-1)y=0
- Use your favorite software - e.g., MATLAB, MAPLE, etc. - to plot
x½J0(x) for x = 0 to 50. (There are
canned programs for computing Bessel functions. In MATLAB, the
function is
besselj
.)
Assignment 21 - Due Thursday, June 30.
- Read section 1.3 of the Notes
on Special Functions.
- Do the following problems.
- Consider the following operators, spaces, and inner products. In each case, show that the operator is self adjoint. All of these require integration by parts. See the example in section 1.3.
- L[f] = f′′, V = {f in C(2)[2,4] | f(2)=0 and f(4) = 0}, < f, g > = ∫24 f(x)g(x)dx.
- L[f] = f′′ − f′, V = {f in C(2)[0,∞) | f(0)=0 and f is "nice" at ∞}, < f, g > = ∫0∞ f(x)g(x)e−xdx.
- L[f] = f′′ + (2/x)f′, V = {f in C(2)[0,∞) | f is "nice" at 0 and f′(1)=0}, < f, g > = ∫01 f(x)g(x)x2dx.
Updated 6/29/06 (fjn).