Math 311-101 Assignments - Summer I, 2008
Assignment 1 - Thursday, May 29.
- 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, May 30.
- 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
w = (2, 1, -1, 1).
- Find the lengths of v and w.
- Find the angle between v and w.
- Find the projection p of v onto the direction
of w.
Assignment 3 - Monday, 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 2.1B (pg. 58): 18
- Section 2.2C (pgs. 70-71): 17, 18, 21
- Section 2.3D (pgs. 80-81): 6, 10, 17, 33
Assignment 4 - Tuesday, June 3.
- Read sections 2.4, 2.5 in the text. Also, read
my The Rank of a
Matrix.
- 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 - Wednesday, June 4.
- Read section 3.1 and 3.2 in the text.
- Do the following problems.
- Section 2.3D (p. 81): 40.
- Section 2.4C (p. 87): 6, 8, 19, 26 (see errors).
- Either find the inverse of A below or show that it doesn't exist.
Assignment 6 - Thursday, June 5.
- Read sections 3.1, 3.2, and 3.3 in the text.
- Do the following problems.
- Section 2.5 (pgs. 98-99): 5, 6, 11
- Use Theorem 5.7 in section 2.5 to determine whether the matrix A
=
1 −2 2
−1 1 −2
2 −3 5
is invertible. If it is, use the algorithm from section 2.4 to the
inverse.
- Section 3.1 (pgs. 110-111): 3, 4, 6
Assignment 7 - Friday, June 6.
- Read sections 3.1 and 3.2 in the text.
- Do the following problems.
- Section 3.1 (pgs. 110-111): 12, 16, 19
- Hand these in with the next
assignment: Section 3.2 (pgs. 118-119): 1, 2
Assignment 8 - Due Tuesday, June 10.
- Read sections 3.3 and 3.4 in the text.
- Do the following problems.
- Section 3.2 (pgs. 118-119): 1, 2, 7, 11, 19, 26
Assignment 9 - Wednesday, June 11.
- Read sections 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 12.
- 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 - Friday, June 13.
- 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 bases for the image of A
(column space), row space of A, and 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 16.
- Section 3.5B (pgs. 137-138): 40, 41(a,b).
- 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 spans the same set as S. Is this set a basis for P2?
Explain.
Assignment 13 - Tuesday, June 17.
- Read sections 3.6B, 3.6C, and 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 a (polynomial) basis for image(L). What is the rank of L?
- Find a (polynomial) basis for null(L). What is the nullity of L?
- Section 3.6A (pg. 148): 5, 6, 17
Assignment 14 - Due Wednesday, June 18.
- Read section 3.7 in the text.
- Do the following problems.
- Section 3.6C (pg. 154-155): 6, 9, 14, 16, 17
- For the matrix in exercise 17 above, solve x′ =
Ax. As we did in class, give your answer in the form U (matrix)
U-1x(0).
Assignment 15 - Thursday, June 19.
- Read section 3.7 in the text.
- Do the following problems.
- Let L : P3 → P3 be the linear operator
defined by L[p] = [(1-x2)p′]′. Find the
eigenvalues and eigenvectors (polynomials) for L.
- Section 3.7A (pg. 158): 2, 11, 13
- Show that yTAx
defines an inner product on R2, where A =
3 1
1 3
-
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|| (Recall ||f|| = (< f, f >)1/2.)
- < f, g >
- The angle between f and g.
Assignment 16 - Due Friday, June 20.
- Read section 3.7 in the text.
- Do the following problems.
- Consider the usual inner product in R4,
< x, y > = yTx.
- Verify that the set B = {(½ ½ ½
½)T, (½ ½ −½
−½)T, (½ −½ ½
−½)T} is orthonormal.
- Leit U = span(B) and let v = (1 2 1
−1)T. Use Theorems 3.7.4 and 3.7.5 to find the
vector umin in U that minimizes ||v
− u|| among all u ∈ U.
- Find the Fourier coefficients a0, a1,
a2, b1, b2 for f(x) = x + 1. See
p. 161 for the definition of these coefficients.
- 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) = |x|.
Assignment 17 - Due Tuesday, June 24.
- 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 25.
- Review Gauss's Theorem and Stokes's Theorem (sections 9.4 and 9.5).
- Do the following problems.
- Consider the sphere of radius a. In class, I used a geometric
argument to show that the standard normal was N = a
sin(φ)r. Calculate N directly from the determinant
form of the cross product. After some algebra, you should get the
same answer.
- For the following surfaces, calculate the standard
normal N, the unit normal n, and the area element
dσ.
- Cylinder. r(θ,z) = a cos(θ)i + a
sin(θ)j + z k.
(Answer: N = a cos(θ)i a
sin(θ)j, n =N/a, and dσ = a
dθdz.)
- Cone. r(r,θ) = rcos(θ)i +
rsin(θ)j + αrk.
(Answer: N = −rα cos(θ)i
−rαsin(θ)j + rk, n
=r−1(1+α2)−½
N and dσ =
r(1+α2)½
drdθ.)
- Paraboloid. r(r,θ) = rcos(θ)i +
rsin(θ)j + αr2 k.
(Answer: not given.)
- Use formula (3.1), pg. 420, and the surace area element for the
sphere of radius a, dσ =
a2sin(φ)dθdφ, to find the sphere's suraface
area.
Assignment 19 - Due Thursday, June 26.
- 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
- Section 9.4 (pgs. 437-438): 5, 7
Assignment 20 - Due Friday, June 27.
- Read sections 9.3 to 9.5.
- 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.
Updated 6/26/08 (fjn).