Rectangular Matrices
Spring 2011
I am very grateful to my student Yara Abdallah who typed these lecture
notes.
Contents
1 Multiplication
2
2 Adjoint
2
3 The eigenvalue decomposition of an arbitrary m n matrix
3.1 Construction and Proof . . . . . . . . . .
Math 5101
HOMEWORK 3
1. Let V = Sp(1, cos t, cos 2t, cos 3t, cos 4t) be the linear span of these ve
linear independent continuous functions over R.
a) Show that cfw_1, cos t, cos2 t, cos3 t, cos4 t is a basis for V .
b) What is dim V ?
2.6 Solve, write up
Homework 4
5101 (AU 2013)
Name(s):
1. a) For the basis dual to the standard basis cfw_e1 , e2 , e3 give explicit
formulas for e1 (x1 , x2 , x3 ), e2 (x1 , x2 , x3 ), e3 (x1 , x2 , x3 ).
b) Write the linear functional f : R3 R dened by
(1)
f (x1 , x2 , x3
Homework 1
5101 (AU 2013)
Name(s):
1. Check that R2 with the usual, component-wise, addition and scalar multiplication is
a vector space over the scalars R.
2. a) Show that the set of continuous functions over the interval [a, b]
C [a, b] = cfw_f : [a, b]
Homework 5
Math 5101 (AU 2013)
Name(s):
1. For each of the following matrices
12
12
1001
A=
, B = 0 0 , C =
48
0101
21
a) Find 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
Math 5101
HOMEWORK 8
1. Verify that the polarization identity
3
ik ik x + y
x, y =
2
k=0
holds in any inner product space (V, , ) over the complex numbers F = C.
2. Consider R3 equipped with the Euclidian inner product: x, y = x y.
Let v = (1, 2, 3) R3 .
5.1 Introduction 251
ination steps produced the exact answer in a nite time. (Or equivalently, Cramers
rule gave an exact formula for the solution.) In the case of eigenvalues, no such
steps and no such formula can exist, or Galois would turn in his grav
Math 5101
Homework 11
1. Consider the quadratic form on R3
F (x) = x2 2x2 + x2 + 4x1 x2 + 8x1 x3 + 4x2 x3
1
2
3
a) Find an orthonormal basis u1 , u2 , u3 , and the numbers 1 , 2 , 3 so that
3
if x =
3
yi ui
2
i yi
then F (x) =
i=1
i=1
b) Find an orthogona
Math 5101
Homework 12
Name(s):
1. If every entry in an orthogonal matrix is either
the matrix?
23
69
2. Write the rank one matrix M =
1
4
or 1 , how big is
4
as xyT and write M T in the
same form.
3. Consider M =
111
,b=
111
0
. Find the pseudoinverse M +
August 24, 2012
LINEAR ALGEBRA
RODICA D. COSTIN
Let us measure everything that is measurable, and
make measurable everything that is not yet so.
Galileo Galilei
Contents
1. Vector spaces
1.1. Notations
1.2. The denition of vector spaces
1.3. Subspaces
1.4
Chapter 3
Linear equations
3.1. The matrix exponential
We begin with the study of the autonomous linear rst-order system
x(t) = Ax(t),
x(0) = x0 ,
(3.1)
where A is an n by n matrix. Here, as usual, we write Ax for the matrix
product whose components are g
September 12, 2012
EIGENVALUES AND EIGENVECTORS
RODICA D. COSTIN
Contents
1. An example: linear dierential equations
2. Eigenvalues and eigenvectors: denition and calculation
2.1. Denitions
2.2. The characteristic equation
2.3. Geometric interpretation of
5101 (AU 2012)
Homework 1
Name(s):
Updated Homework policy: if you work in groups for your homework assignments
(up to three students in a group) please submit only one write-up, signed by all members
of the group. All co-signers will receive the same gra
Math 5101
Due Friday Sept 14
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 poly
Math 5101
Due Friday Sept 21
HOMEWORK 3
Solve, write up solutions and hand in for grading the
problems 2.6.1, 2.6.2, 2.6.5, 2.6.7, 2.6.13, 2.6.16 posted on
Carmen,
in the module II. Problems from Strangs book, 3rd ed.,
topic Problems Ch2.
Solve more probl
Math 5101
Whenever you use symbolic calculation software attach a print out and
explain what you calculated and why. For Problem 4. also include the signicance of the columns of the transition matrix obtained.
HOMEWORK 4
1. For each of the following matri
Math 5101
Due Friday Oct 5
HOMEWORK 5
Solve, write up solutions and hand in for grading
all problems from 5.1.1 to 5.1.6,
also 5.1.7, 5.1.11, 5.1.13, 5.1.14, 5.1.15, 5.1.17, 5.1.18. posted on
Carmen,
in the module II. Problems from Strangs book, 3rd ed.,
Math 5101
HOMEWORK 7
1. Recall the complex roots of 1:
a) Show that the n complex solutions of the equation z n = 1 are
zk = exp[2ik/n], k = 0, 1, . . . , n 1 (called the nth roots of unity).
k
b) Show that zk = z1 . (z1 is called the primitive nth root o
Math 5101
Homework 11
due Friday Nov. 30, 2012
1. Show that the pseudoinverse of a vector x Cn is
x+ =
0,
1
x
if x = 0
x , if x = 0
2
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. Let u1 , . . . , ur
September 9, 2012
LINEAR TRANSFORMATIONS
RODICA D. COSTIN
Contents
2. Linear Transformations
2.1. Denition and examples
2.2. The matrix of a linear transformation
2.3. Operations with linear transformations
2.4. Null space and range
2.5. The rank of a mat
Math 5101
HOMEWORK 9
1. Recall the complex roots of 1.
a) Show that the n complex solutions of the equation z n = 1, called the nth
roots of unity, are zk = exp[2ik/n], k = 0, 1, . . . , n 1.
k
b) z1 is called the primitive nth root of unity. Show that zk