MATH 601, AUTUMN 2008
Additional homework I, October 2
PROBLEMS
1. Determine if the set X consisting of the following three functions t 1, t2 + 3 , t t2
of the real variable t , spans the whole real vector space V of real-valued polynomials in t
Math 602 (2011)
Homework 3
1. Let A be a self-adjoint matrix. Show that there is a self-adjoint matrix B
so that
B 3 = A(A 2I )(A 3I )
What are the eigenvalues of B in terms of the eigenvalues of A? Will B
commute with A?
2. Assume that A is a symmetric m
Math 601
HOMEWORK 4
1. For each of the following matrices
12
1001
12
A=
, B = 0 0 , C =
48
0101
21
a) nd the null space.
b) Find the column space.
c) Is the matrix invertible? If so, nd its inverse, and check your answer.
d) Does the matrix have a left in
Math 602
Homework 4
1. Consider the quadratic form on R3
x2 2x2 + x2 + 4x1 x2 + 8x1 x3 + 4x2 x3 = F (x)
1
2
3
Find the following:
a) an orthonormal basis of unit vectors cfw_e1 , e2 , e3 , and a set cfw_1 , 2 , 3
so that
3
3
2
i yi
yi ei F (x) =
x=
1
1
3
Math 601
HOMEWORK 6
Solve the problems 1., 2. below, and also from Strang, 3rd ed., problems
5.2.1, 5.2.3, 5.2.5, 5.2.6, 5.2.7 on p. 260-261 (see the .pdf le handed out in
connection to HW 5).
1. Consider an n n matrix M with entries in F ( = R or C) with
Math 602
Homework 6
1. Show that the pseudoinverse of a vector x Cn is
x+ =
0,
1
x
2
if x = 0
x , if x = 0
2. Find the SVD and the pseudoinverse of the following matrices:
M=
1
1
2 2
L=
10 2 10 2
5 11 5 11
3. The null space of a matrix is also called its
Math 601
HOMEWORK 7
Solve the problems 1., 2. below, and also from Strang, 3rd ed., problems
3.1.9(use any method), 3.1.12, 3.1.16, 3.1.19 on p.142, and 3.2.3, 3.2.10,
3.2.12 on p.151 (see the new .pdf le handed out).
1. Let (V, , , ) be an inner product
Math 602
Homework 7
Please note the following very useful inequalities:
|f (t)| f
for all t S if f
= sup |f (x)|
xS
and
b
b
f (t) dt
a
f (t) dt
for a < b
a
1. Consider the sequence of functions fn (x) = xn .
a) Show that the sequence fn is point-wise con
October 15, 2010
EIGENVALUES AND EIGENVECTORS
RODICA D. COSTIN
Contents
1.
An example: linear dierential equations
1
1. An example: linear differential equations
One important motivation for the study of eigenvalues is solving linear
dierential equations.
Math 601
HOMEWORK 4
1. For each of the following matrices
12
1001
12
A=
, B = 0 0 , C =
48
0101
21
a) nd the null space of A (recall = the null space of the linear transformation x Ax).
b) Find the column space of A (recall = the range of the linear trans
October 13, 2010
LINEAR TRANSFORMATIONS
RODICA D. COSTIN
Contents
2. Linear Transformations
2.1. Denition and examples
2.2. The matrix of a linear transformation
2.3. Null space and range
2.4. Dimensionality properties
2.5. Exercises
2.6. Invertible trans
MATHEMATICS 601
Mathematical Principles in Science I
Au 2010
Info for the sections at 1:30 p.m., 2:30 p.m. MWF
Instructor: Dr. Rodica D. Costin,
Oce: 436 Math Tower
Oce hours: MF 12:30 - 1:30 p.m. or by appointment
e-mail: rcostin@math.ohio-state.edu
web
November 24, 2010
NORMED SPACES, THE DUAL SPACE
RODICA D. COSTIN
Contents
1. The Triangle Inequality
2. Normed spaces
2.1. Examples in nite dimensions.
2.2. Examples in innite dimensions.
3. The norm of a matrix
1
2
2
3
3
1. The Triangle Inequality
Let (V
Math 601
HOMEWORK 2
1. Show that the set of 22 matrices with real entries, with usual addition
and multiplication with scalars, is a real vector space. Find its dimension
and a basis.
2. For each n denote by Pn the vector space of polynomials of degree at
Math 602
HOMEWORK 1
1. (a) Show that the product of two unitary matrices is also a unitary
matrix.
(b) If U is unitary, is its inverse U 1 unitary?
(c) Is the identity matrix unitary?
2.
(a) Show that the determinant of a unitary matrix has absolute value
Math 601
HOMEWORK 1
1. Decide which of the following sets are subspaces in R3 , and for each
subspace nd a basis, its dimension and describe the type of geometrical
gure it is.
U1 = cfw_(x1 , x2 , x3 ) | x1 = 2t, x2 = t, x3 = 5t, t R
U2 = cfw_(x1 , x2 , x
MATH 601, AUTUMN 2008
Additional homework III, October 18
PROBLEMS
1. Determine dim T (R3 ) for the linear 1 x y = 2 T 1 z 1
operator T : R3 R4 given by 1 1 x 0 6 y , 33 z 15
with vectors of R3 and R4 written in the column form. 2. For
MATH 601, AUTUMN 2008
Additional homework V, PROBLEMS October 30
1.
Find the solution v(t) = (x(t), y(t), z(t) to the system of ordinary dierential v = Tv
equations
with the initial condition v(0) = (1, 1, 1), where T : R3 R3 is the operator
MATH 601, AUTUMN 2008
Additional homework VII, PROBLEMS November 19
1. Let V be an inner-product space of dimension n, and let v be a nonzero vector in V . Use the rank-nullity theorem to show that dim v = n 1. 2. Let W be a subspace of an inner-p
MATH 601, AUTUMN 2008
Additional homework VIII, PROBLEMS 1. Let , be the symmetric bilinear form on R2 given by (x, y), (a, b) = 4xa + 2yb xb ya. Verify that , is an inner product and nd the minimum distance d between the vector v = (1, 0) and all
MATH 601, AUTUMN 2008
The index notation with the summing convention
The index notation includes the following conventions: (a) In each term (monomial) forming a given expression, any index (that is, a subscript or superscript) may appear at most tw