Math 311-501 Assignments
Assignment 1 - Due Thursday, 25 January 2007
- Read sections 2.1-2.3 in the text.
- Do the following problems.
- Section 1.1: 9, 29
- Section 1.2: 30(a)
- Section 1.3: 18, 19
- Section 1.4: 21, 23
- Section 1.6: 9, 16
- Additional 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 2 - Due Thursday, 1 February 2007
- 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
- Section 2.2D (p. 73): 2, 11 (see errors)
- Additional problems
- For the matrices R, S, u, v below, find the required combinations, 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 below. 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, the particular solution, and the solution to the
corresponding homogeneous system.
-
Assignment 3 - Due Thursday, 8 February 2007
- Read sections 2.4, 2.5 and 3.1A in the text.
- Do the following problems.
- 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): 39, 49.
- Section 2.4C (p. 87): 7, 11, 24, 30 (see errors), 33.
- Either find the inverse of A below or show that it doesn't exist.
Assignment 4 - Due Thursday, 15 February 2007
- Read sections 3.1 and 3.2 in the text.
- Do the following problems.
- Section 2.5 (pg. 98): 1, 5, 6. 7, 9
- Let c be a scalar and A an n×n matrix. Use the properties
of determinants to show that det(cA)=cndet(A).
- Theorem 2.5.7 in the text states that A is invertible if and only
if det(A) ≠ 0. Use this for the following problems:
Section 2.5 (pg. 99): 21, 22, 23
Assignment 5 - Due Thursday, 22 February 2007
- Read sections 3.1 and 3.2 in the text.
- Do the following problems.
- Section 2.5 (pg. 99): 15, 18 (Cramer's rule.)
- Section 3.1 (pg. 110): 3, 5, 7 (In each case, find the matrix for
the linear function f.)
Assignment 6 - Due Thursday, 1 March 2007
- Read sections 3.2 and 3.3 in the text.
- Do the following problems.
- Section 3.1 (pg. 110): 12, 16, 19, 23
- In Section 3.2, for the spaces Mm,n ,
Pn, C[a,b], C(k)[a,b], write out the
definition of each space, and definitions of ``vector'' addition and
of multiplication by a scalar in each space. Take m = 3, n = 4, k = 2,
a = 3/2, b = 5/2. For these values, give particular examples of
``vectors'' in each space.
Assignment 7 - Due Thursday, 8 March 2007
- Read sections 3.4 and 3.5 in the text.
- Do the following problems.
- Section 3.1 (pg. 110): 33
- Section 3.2 (pgs. 118-119): 7, 8, 11, 13, 16, 22, 26
- Section 3.4 (pgs. 130-131): 9, 10
Assignment 8 - Due Thursday, 22 March 2007
- Read sections 3.5 and 3.6 in the text, and my notes, Methods of Finding Bases
- Do the following problems.
- Section 3.4 (pgs. 130-131): 13, 21(a,b,c)
- Section 3.5B (pgs. 137-138): 7, 10, 13, 15, 20, 23
Assignment 9 - Due Tuesday, 2 April 2007
- Read sections 3.5 and 3.6 in the text, and my notes, Methods
of Finding Bases
- Do the following problems.
- 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?
- 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 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.
- Section 3.6A (pg. 148): 6, 8, 17
Assignment 10 - Due Thursday, 19 April 2007
- Read sections 3.7A and 3.7B in the text.
- Do the following problems.
- Section 3.7A (pg. 158): 13
-
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.
-
Find the (discrete) least-squares straight line fit to the data below
t = 0, 1, 2, 3; y = 1.1, 2.9, 5.0, 6.9
- Find the polynomial p(x) = a0 + a1x +
a2x2 that gives the (continuous) least-squares fit
to y = e-x in the inner product < f,g > =
∫01 f(x)g(x)dx.
- Let A be a real n×n matrix that is also symmetric; that is,
A=AT. In addition, consider the standard inner product on
Rn, < x, y > =
yTx.
-
Show that < Ax, y > = < x, Ay >
-
You are given that the eigenvalues and eigenvectors of A are all
real. Show that eigenvectors corresponding to distinct
eigenvalues are orthogonal in the inner product < x, y >.
- 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
- 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 11 - Due Thursday, 26 April 2007
- Do the following problems.
- Section 3.5B (pgs. 137-138): 17
- Review problems, chapter 3 (p. 172): 22
- Let L : P2 → P2 be defined by L[p]=
(x2 - x + 3)p'' − (4x − 1)p' + 6p.
- Show that L is linear.
- Find the matrix A of L relative to the standard basis B = {1, x,
x2}.
- Find bases for the 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.
- In the diagram for the spring-mass system in problem 6 of Test 2,
use 4k (rather than 2k) for the middle spring's constant. Find the
normal modes of this system, given that Newton's laws applied to the
system give these equations of motion:
- In class we discussed the Gram matrix G for a least squares
problem. Given a linearly independent set of vectors
{w1,..., wn} and an inner product
< , >, we define the entries of G via
Gj,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, cos(x), sin(x)}, < f,g > =
∫-ππ f(x)g(x)dx
- Use the Gram matrix and the inner product from (b) above to do a
least-squares fit for f(x) = π − |x| on [−π,&pi]
using a trigonometric polynomial T(x) = a0 +
a1cos(x) + b1sin(x).
Updated 4/22/07 (fjn).