Homework Set 1 for MAS 4105
The following problems are due on Friday, January 20:
1. Determine whether the set V = R2 with the indicated operations is
a vector space over R.
( x1 , x2 ) + ( y1 , y2 )
Answers for Practice Problem Set 1
00
00
01
10
,
,
,
01
10
00
00
(b) cfw_(1, 1, 0), (5, 0, 1)
(c) cfw_X, X 2
1. (a)
2. (a) W is a subspace of R3 .
(b) W is not a subspace of R3 , since (0, 1, 0) W bu
MAS 4105Practice Problems for Exam #1
1. Find a basis for each vector space V . (No justication needed for this problem.)
(a) V = M22 (F )
(b) V = cfw_(x1 , x2 , x3 ) R3 : x1 + x2 5x3 = 0
(c) V = cfw_
Foreword
There are a zillion books on linear algebra, yet this one nds its own unique place
among them. It starts as an introduction at undergraduate level, covers the essen-
tial results at postgradu
xii Contents
6.2.2 Denition of the Tensor Product of Unitary Spaces,
in Analogy with the Previous Example . 249
6.3 Matrix Representation of the Tensor Product of Unitary Spaces . . . . 250
6.4 Multip
Acknowledgements
Thanks are due to several people who have helped in various ways to bring Professor
Vujicics manuscript to publication. Vladislav Pavlovit': produced the initial Latex
copy, and subse
x Contents
2 Linear Mappings and Linear Systems . 59
2.1 A Short Plan for the First 5 Sections of Chapter 2 . 59
2.2 Some General Statements about Mapping . 60
2.3 The Denition of Linear Mappings (Lin
Exercises 23
7.
10.
Suppose A is an upper triangular matrix. Show that A1 exists if and only if all
elements of the main diagonal are non zero. Is it true that A1 will also be upper
triangular? Explai
Exercises 21
40. Remember the Coriolis force was 29 X VB where Q was a particular vector which
came from the matrix Q (t) as described above. Show that
3W?) ' i (150) 5(3) 4&0) km *th)
Q (3) = i) 'j (
22
Exercises
4 3 2
(b) 1 7 8 (The answer is 375.)
3 9 3
1 2 3 2
1 3 2 3 .
(o) 4 1 5 0 , (The answer 1s 2.)
1 2 1 2
2. If A1 exist, what is the relationship between det (A) and det(A_1) . Explain your
18 Exercises
35. TSuppose you have nitely many linear mappings L1, L2, - - * ,Lm which map V to V
where V is a subspace of IF and suppose they commute. That is, 1,115, : L, L, for all
i, 3'. Also supp
Exercises
12. Let x = (1, 1, 1) and y = (0, 1, 2) . Find xTy and xyT if possible.
1 0 12
xTy: 1 (012): 0 12
1 0 1 2
0
xyT:(-1m1 1) 1 :1
2
13. Give an example of matrices, A, B, C such that B 75 C, A 7
12
Exercises
(cosh/3) sin('n'/3) ) ( cos(7r/4) sin(s'/4)
sinCs/S) cosh/3) sin(s/4) cos('n'/4)
_ (ax/amass -%\/\/)
warg 2 yams/i
. Find the matrix for the linear transformation which rotates every vect
Exercises 25
e U U
0 at cost etsint
0 etcostmetsint etcost+e sint
U 0
(cos 13 + sin t) m (sin t) et
e (cost sin t) (cost) et
14. Let A be an r >< r matrix and let B be an m x m matrix such that r + m
2 1 Vector Spaces
The vector spaces R" for a : 2, 3, play an important role in geometry, describing
lines and planes, as well as the area of triangles and parallelograms and the volume
of a parallelep
vi Foreword
III-vi
December 2005.
Having known Milan well, as my thesis advisor, a colleague and a dear friend, I
am certain that he would wish this book to be dedicated to his wife Radrnila and his
s
10 1 Vector Spaces
3. It is associative with respect to scalar multiplication Md >< 3) : (hi) *5 : - (
For k :> 0 it is obvious, since kri and RE? have the same direction as ti an
respectively.
k").
d
g 1 Vector Spaces
(c+d)a= (as) [j] = [(5:35] = [313;] = [3;] + [3;] =ca+da;
(ii) The associative property of the multiplication of numbers with respect to scalar
multiplication:
(cdhi = (ca) [ 2 [3;]
1.3 Vectors in a Cartesian (Analytic) Plane R2 5
We simply add one vector after another. This is the thi rd property.
Each vector a has its unique negative & (the additive inverse), which has the
sa
4 1 Vector Spaces
Now. connect the initial point of the rst vector to the terminal point of the second
vector (we add the second vector to the rst one). This is the triangle rule for the
addition of
6 1 Vector Spaces
Thus, each rectangular coordinate system transforms the Euclidean plane into a
Cartesian (analytic) plane. Analytic geometry is generally considered to have been
founded by the Frenc
1.4 Scalar Multiplication (The Product of a Number with a Vector) 7
Since the components are real numbers, and the addition of real numbers makes
R an Abelian group, we immediately see that R2 is also
1.2 Geometrical Vectors in a Plane 3
translation
This relation in the set of all vectors in the plane is obviously reexive, symmetric
and transitive (an equivalence relaiiaanerify), so it produces a
Chapter 1
Vector Spaces
1.1 Introduction
The idea of a vector is one of the greatest contributions to mathematics, which came
directly from physics. Namely, vectors are basic mathematical objects of c
Contents
1 Vector Spaces . 1
1 . 1 Introduction . 1
1.2 Geometrical Vectors in a Plane . 2
1.3 Vectors in a Cartesian (Analytic) Plane 1&2 . 5
1.4 Scalar Multiplication (The Product of a Number with a
Contents xi
4
Dual Spaces and the Change of Basis . 145
4.1 The Dual Space U; of a Unitary Space U . 145
4.2 The Adjoint Operator . 153
4.3 The Change of Bases in V.I (F) . 157
4.3.1 The Change of the
1.5 The Dot Product of Two Vectors (or the Euclidean Inner Product of Two Vectors in R3) 9
If we choose two unit vectors (orts) iand f in the directions of the x and y axes,
respectively, then we imme
Author Editor
Milan Vujicic Jeffrey Sanderson
(19312005) Emeritus Professor,
School of Mathematics & Statistics,
University of St Andrews,
St Andrews,
Scotland
ISBN: 9783540746379 eISBN: 9783540-74639
26 Exercises
F. 19 Exercises
3.6
1. Let m < n and let A be an m X n matrix. Show that A is not one to one. Hint:
Consider the n. x a matrix A1 which is of the form
Alas)
Where the 0 denotes an (n. w m
24
Exercises
Use Laplace expansion and the rst part to verify F (t) =
( it" (t) 5" (t) 0"(15) ) ( a (t) b (t) 60*) )
det d(t) e (t) f (t) + det d (t) e" (t) f" (t)
g (t) h (t) W) 9 (t) h (t) W)
( a (t
Exercises 15
27. TSuppose V is a subspace of F and T : V > lFP is a nonzero linear transformation.
Show that there exists a basis for Im (T) E T(V)
cfw_Tv1,-~,Tvm
and that in this situation,
cfw_v1 a
16
Exercises
30. TLet A be a linear transformation from V to W and let B be a linear transformation
31.
from W to U where 1/: W, U are all subspaces of some lFP. Explain why
A (her (BA) g ker (B), ker