Math 311-102 Assignments - Summer I, 2007
Assignment 1 - Thursday, May 31.
- Read sections 1.1-1.4, 1.6 in the text.
- Do the following problems.
- Section 1.1: 9, 10, 12, 24
- Section 1.2: 30(a)
- Section 1.3: 6(a,b,c)
Assignment 2 - Friday, June 1.
- Read sections 2.1A, 2.2 in the text. Also, read my Notes on Row Reduction.
- Do the following problems.
- Section 1.3: 18
- Section 1.4: 23
- Section 1.5: 27
- Section 1.6: 9
- Section 1.6: 16
- 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 Monday, June 4.
- 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 - Tuesday, June 5.
- Read sections 2.5, 2.5 in the text.
- Do the following problems.
- Section 2.2D (p. 73): 2, 4, 11 (see errors)
- Consider the matrix A =
1 −3 2 −2 2
−1 3 −2 1 −3
2 −6 5 −3 5
Find the reduced echelon form of A, the rank of A, the nullity of A,
and the leading columns of A. Determine whether the columns of A are
LI or LD. Solve Ax = 0.
-
Consider the set of vectors S below.
S ={(1 0 2 −1 3)T, (1 −2 1 1 0)T,
(−1 1 0 2 −1)T}
- Determine whether S is LI or LD.
- Can v = (4 −5 4 −1 4)T be
written as a linear combination of vectors in S? Are the coefficients
in the linear combination unique?
Assignment 5 - Due Wednesday, June 6.
- Read sections 3.1 and 3.2 in the text.
- Do the following problems.
- Section 2.3D (p. 81): 40.
- Section 2.4C (p. 87): 8, 26 (see errors).
- Either find the inverse of A below or show that it doesn't exist.
- Section 2.5 (pgs. 98-99): 5, 6.
Assignment 7 - Due Friday, June 8.
- Read sections 3.2 and 3.3 in the text.
- Do the following problems.
- Section 3.1 (pgs. 110-111): 12, 16, 19
- Section 3.2 (pgs. 118-119): 7, 11, 16, 18, 26
Assignment 8 - Due Tuesday, June 12.
- Read sections 3.3 and 3.4 in the text.
- Do the following problems.
- Section 3.2 (pgs. 118-119): 20, 25, 27
- Let A be an m×n matrix. Show that the set of all x
in Rn such that Ax = 0 is a subspace
of Rn. (Hint: you have done this problem in disguise
once before.)
Assignment 9 - Due Wednesday, June 13.
- Read section 3.5 in the text.
- Do the following problems.
- Section 3.3 (pgs. 125-126): 17(a,b), 20, 25
- Section 3.4 (pgs. 130-131): 9, 10, 13
Assignment 10 - Due Thursday, June 14.
- Read section 3.5 in the text and my notes,
Coordinate Vectors and Examples.
- Do the following problems.
- Section 3.4 (pgs. 130-131): 21(a,b)
- Determine whether {1, cos(2x), cos2(x)} is LD or
LI. (Hint: Use a trig identity.)
- Let u1 = i + j -k,
u2 = 3i - j +k,
u3 = j + k. Find
[u1]B,
[u2]B, and
[u3]B. Use these coordinate vectors to
show that B = {u1, u2,
u3} is a basis for 3D space.
- Show that C = {1 − 2x, 1 + 2x, 1 − x2} is a
basis for P2. Find [(2x −
1)2]C.
- The set of functions C={cosh(x), sinh(x)} is a basis for the
space of homogeneous solutions to y′′ - y = 0. Show that
for any constant α the function y=cosh(x+α) solves
y′′ - y = 0. Find [cosh(x+α)]C. (This
amounts to an identity for the hyperbolic functions similar to one for
the trig functions.)
Assignment 11 - Due Friday, June 15.
- 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 12 - Due Monday, June 18.
- Read section 3.6 in the text.
- Do the following problems.
- Section 3.5B (pgs. 137-138): 40, 41(a,b). Also, in problem 41, find
the matrix of S in the basis B={1,x,x2}.
- 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.
- Consider the set S = {1-x,2-2x,x2 + x,
x2+x+1, 1+x} for P2. Without any calculation,
explain why S is LD. Find a linearly independent set of vectors in S
that spns the same set as S. Is this set a basis for P2?
Explain.
Assignment 13 - Due Tuesday, June 19.
- Read section 3.7 in the text.
- Do the following problems.
- Let L : P2 → P2 be defined by L[p] =
(x2 + 2)p'' + (x-1)p' - 3p.
- 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}.
- Section 3.6A (pg. 148): 5, 6, 17
Assignment 14 - Due Wednesday, 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 Thursday, June 21.
- Read section 3.7 in the text.
- Do the following problems.
- Section 3.7A (pg. 158): 2, 3, 11
-
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.
Assignment 16 - Due Friday, June 22.
- Read sections 3.7 in the text.
- Do the following problems.
- Find the Fourier coefficients for each of the following functions
that are defined on [−π,π].
- 2x-1
- π - |x|
- x2
- Consider the inner product < f,g > =
∫-11 f(x)g(x)dx. The three polynomials below
form an orthonormal basis for P2 with respect to
this inner product:
p0(x) = 2-½, p1(x) =
(3/2)½x, and p2(x) =
(5/8)½(3x2 − 1).
Use this orthonormal set in conjunction with Theorem 3.7.5 (p. 162) to
find the least squares quadratic fit for f(x) = e− x.
Assignment 17 - Due Tuesday, June 26.
- Review surface integrals (section 9.3).
- 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, cos(x), sin(x)}, < f,g > =
∫0π f(x)g(x)dx (Note: the lower limit is
0, not π.)
- 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 Wednesday, June 27.
- Review Gauss's Theorem and Stokes's Theorem (sections 9.4 and 9.5).
- Do the following problems.
- Section 9.3 (pgs. 429-430): 5, 6, 7, 9
Assignment 19 - Due Thursday, June 28.
- Stokes's Theorem (section 9.5). Read the Notes
on Special Functions through section 1.2 (pg. 5).
- Do the following problems.
- Section 9.4 (pgs. 437-438): 5, 7, 9, 17
Assignment 20 - Not to be turned in.
- Read the Notes
on Special Functions through section 1.2 (pg. 5).
- Do the following problems.
- Section 9.5 (pgs. 447-449): 7, 9
- Verify Stokes's Theorem in case F= 2yi + 3xj
− z3k and S is the upper hemisphere of
x2 + y2 + z2 = 4, which has the
circle x2 + y2 = 4, z = 0, as a boundary. Use
the normal with positive z component.
- 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
Updated 6/28/07 (fjn).