Hello Again, Linear Algebra: A Second Look At The Subject Through A Collection Of Exercises And Solutions

98 Pages·2014·4 MB·English

by  Pavel Grinfeld

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Introduction
Chapter 1
Linear algebra is a grand subject. Because it is fundamentally different from
any high school mathematics, and because of the wildly varying quality of
instructors, not all students enjoy learning it. If you hated it, I blame the
instructor. There I said it. In either case, whether you loved it or hated it, it takes
several passes to learn linear algebra to the point that it becomes one of your
favorite tools, one of your favorite ways of thinking about practical problems.
This little textbook invites you on your second pass at linear algebra. Of
course, your second pass may take place alongside your first pass. You may find
this textbook particularly useful if you are studying for a test. Our goal is to take
a step back from the mechanics of the subject with an eye towards gaining a
larger view. A larger view, however, is achieved in small steps. We are not
hoping for a big revelation but for a few small aha! moments. It simply takes
time to put together the grand puzzle of linear algebra. You will get there, and
the point, as the cliche goes, is to enjoy the ride.
Psychology in mathematics is everything. I chose the topics according to the
impact I feel they would make on your relationship with linear algebra. The
textbook's utmost goal is to make you feel positively about the subject. You will
find that some topics are surprisingly simple, others surprisingly tough. Some
topics have important applications, others have none at all. Some were well
presented in your linear algebra course, others skipped altogether. However, I
hope you will find that all topics bring you a little closer to the subject of linear
algebra.
There are several points that this textbook emphasizes over and over again.
Some of these points are made explicitly. Others are more difficult to put into
words and it is my hope that these points become clear as a result of considering
the many examples analyzed here. Therefore, I would like to make these points
right here with minimal explanations.
1. The applications of linear algebra are extraordinarily broad. The subject
applies to any type of objects that can be added together and multiplied by
numbers. These objects are united by the term vector and include pointed
segments, polygons or – more generally – functions, sets of numbers organized
in a column, matrices, currents, voltages, substances in chemical reactions,
strains and stresses in elastic bodies, sound signals, portfolios of financial
securities. Need I go on?
2. Linear algebra studies the commonalities among these objects. However,
focusing on the commonalities is counterproductive if one ignores the
differences. It is that the fact that these objects are all uniquely different that
makes linear algebra such a remarkable tool. We therefore invite the reader to
accept and treat each object on its own terms. If the problem relates to pointed
segments which we call geometric vectors, by all means let us have a geometric
discussion and talk about straight lines, lengths, and angles. Do not rush to
associate a geometric vector in the plane with a pair of numbers. It is not
necessary and, in fact, detrimental to clarity, to do so. Unfortunately, just about
every textbook in the land does this and in doing so obscures much of the
psychology of linear algebra. Similarly, when dealing with , lets talk about
entries, rather than lengths and angles. Sets of numbers have entries and nothing
else. Sets of numbers don't have lengths and directions, and when they do
eventually acquire these notions, it doesn't happen in the corrupted way
presented in most textbooks. Finally, functions are functions, and polynomials
are polynomials. There's no need to think of polynomials as "vectors" of
coefficients. Polynomials are cool on their own terms. Other than adding them
and multiplying them by scalars, you can differentiate them and even dilate
them. Can you do that with "vectors" of coefficients?
3. Let the problem dictate the basis, or say no to Cartesian coordinates! This
point is related to the preceding point and applies to more than the subject of
linear algebra. There seems to be a lamentable tradition to refer the space of our
everyday physical experience to Cartesian coordinates, whether or not it is the
best coordinate system and, even more lamentably, whether or not coordinates
are needed in the first place. If I may speak for Descartes, he would have been
appalled by this state of affairs! His ingenious invention of coordinates was
meant to unite the worlds of algebra and geometry. The knee-jerk reaction to
translate a physical problem to numbers, typically by introducing a Cartesian
coordinate systems, and often doing so tacitly, usually leads to loss of simplicity,
vision, clarity, and understanding.
In linear algebra, choosing a coordinate system is equivalent to choosing a
basis and the Cartesian syndrome manifests itself by choosing the orthonormal
basis in the plane:
or the basis for the space of cubic polynomials:
or the basis in :

First of all, as was just discussed in point 2., don't resort to a basis in the first
place, thereby converting a "real-life" problem to numbers, unless the reasons for
doing so are clear to you. Good reasons for doing so, for instance, are described
in Chapter With Or Without a Basis. Secondly, when the time is right to convert
the problem to by decomposing all elements of the problem with respect to a
basis, let the problem dictate your choice of basis.
This is much more than a matter of practicality. If a basis is chosen poorly or,
worse yet, introduced unnecessarily, the artifacts of the basis will obscure the
analysis of the problem. My first linear algebra professor Bill Thurston states in
his famous essay On Proof And Progress In Mathematics that the value
understanding is greater than that of the formal correctness of a solution. For an
engineer (even if you are not an engineer, I'll think of you as an engineer: you
want to create things, don't you?) the value of understanding is even greater
because the products of your labor are required to work. So if don't wish for the
bridge you built to collapse – second guess yourself when you instinctively reach
for a basis when presented with a problem. Heal yourself of the Cartesian
syndrome. Here! – I included a portrait of Descartes, because I think that he
would agree with me.
One too many!
Chapter 2
The figure below shows various arrangements of three vectors in the plane.
Which arrangement has three linearly independent vectors?

In our linear algebra course, we come to this question in the second week.
During the first week, we discuss linear combinations of vectors and the concept
of decomposition: representing vectors by linear combinations of other vectors.
Then comes the definition of linear independence: a set of vectors is linearly
independent if none of the vectors can be represented as a linear combination of
the rest. Linear dependence is the opposite of linear independence: a set of
vectors is linearly dependent if at least one of the vectors can be represented as
a linear combination of the rest.
If you don't feel that the definition of linear dependence is obvious or, at
least, natural in some way – I'm with you! However, this definition has shown
over the years to lead to a very effective way of organizing the subject of linear
algebra. This problem, which culminates in a fundamental realization, illustrates
one aspect of the effectiveness of this definition.
Let us start eliminating the linearly dependent sets. First of all, a linearly
independent set cannot contain the zero vector , since it can be expressed as a
linear combination of any set of vectors! Just set the coefficient to for every
vector in the linear combination. For example, for any two vectors and ,
This linear combination, all of whose coefficients are zero, is called the trivial
linear combination. It is a very special combination and it is perfectly valid. The
definition of linear dependence does not call for a nontrivial linear
combination: just one way – any one way – of expressing one of the vectors in
terms of the rest.
It is also unimportant that the vector cannot be expressed in terms of and
. The definition of linear dependence only requires that there is a vector in the
set that can be expressed in terms of the rest. There is no requirement that every
vector be expressible in terms of the rest.
And so, the arrangements and are out. What about the arrangement ?
The vector certainly cannot be expressed in terms of and , since and
point along the same line and any linear combination of and necessarily
points along the same straight line. However, is a multiple of
and it is therefore a linear combination of the vectors and , and the vector is
not even needed to express . That is not particularly relevant since the
definition of linear dependence does not require that all of the remaining vectors
are used in the linear combination. And even if it did, we could simply assign a
coefficient of to :

This eliminates the arrangement . Similarly, the vectors in the arrangement
are linearly dependent since

This leaves us with the arrangements and which, in a manner of speaking,
are the most linearly dependent of them all! We say this because each of the
vectors can be expressed as a linear combination of rest. (Not that the definition
of linear dependence requires that every vector be expressed as a linear
combination of the rest. Just one of the vectors is enough for linear dependence.)
The vectors in the arrangement are such that
Equivalently,
and
The vectors in the arrangement are such that

A moment of reflection. We owe ourselves a moment of reflection. We
tried, tried, and tried again, but to no avail! No matter what configuration of
three vectors in the plane we try, we end up with a linearly dependent set. What
is going on here? What we are experiencing here is a fundamental law of life: a
set of more than two vectors in the plane is necessarily linearly dependent. The
linear dependence may manifest itself in a variety of ways, as we just
discovered. But the fact of linear dependence is inescapable.
This fact is not an axiom or a postulate. This fact it is not a theorem or a
corollary. This fact is something that slowly crystallized out of mathematical
human experience and gave the geometric inspiration to the concept of linear
dependence on which the subject of linear algebra is built.
In the formal subject of linear algebra, there is a theorem that captures this
experience. The theorem states that all bases in a linear space have the same
number of elements. That common number is called the dimension of the space.
(The dimension of the plane is .) It then follows that any set that contains more
vectors than the dimension of the space is linearly dependent. With a little bit of
formal thinking you will be able to explain why.
Exercise 1
Explain why augmenting a linearly dependent set with an
additional vector cannot result in a linearly independent set.
Solution
If one of the vectors in the original set can be expressed as a
linear combination of the rest, for example,
then the addition of a vector , does not alter the original relationship, which can
now be written as
Exercise 2
Explore the three dimensional case and convince yourself
that four or more vectors are necessarily linearly dependent.
Exercise 3
Show that the polynomials
are linearly dependent. This exercise illustrates the unique power of linear
algebra: you may not be able to determine the linear combination that
expresses one of these polynomials in terms of the rest, but one can state with
confidence that a relationship like that exists.
Solution
Cubic polynomials form a four dimensional set. This is
because the set can serve as a basis. Therefore, being
five vectors in a four dimensional space, the polynomials , ,
, , and are linearly dependent.
Exercise 4
Show that the set of vectors
is linearly dependent.
Solution
Since is three dimensional (explain why), the four given
vectors are linearly dependent.
Exercise 5
Show that the set of vectors
is linearly dependent.