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A Brief Review of Linear Algebra
Algebra is generous; she often gives more than is asked of her.
—Jean-Baptiste le Rond d’Alembert (1717–1783)
Linear algebra is a beautiful subject, almost as beautiful as group theory. But who is
comparing?
I wrote this originally as a review for those readers who desire to learn group theory
but who need to be reminded of some key results in linear algebra. But then the material
grew, partly because I want to have a leisurely explanation of how the basic concepts of
matrix and determinant arise naturally. I particularly want to give a step-by-step derivation
of Cramer’s formula for the matrix inverse rather than to plop it down from the sky. So
then in the end, I decided to put this review at the beginning.
This is of course not meant to be a complete treatment of linear algebra.
∗
Rather, we
will focus on those aspects needed for group theory.
Coupled linear equations
As a kid, I had a run-in with the “chicken and rabbit problem.” Perhaps some readers had
also? A farmer has x chickens and y rabbits in a cage. He counted 7 heads and 22 legs.
How many chickens and rabbits are in the cage? I was puzzled. Why doesn’t the farmer
simply count the chickens and rabbits separately? Is this why crazy farmers have to learn
linear algebra?
In fact, linear algebra is by all accounts one of the most beautiful subjects in mathemat-
ics, full of elegant theorems, contrary to what I thought in my tender years. Here I will
take an exceedingly low-brow approach, eschewing generalities for specific examples and
building up the requisite structure step by step.
∗
Clearly, critics and other such individuals looking for mathematical rigor should also look elsewhere. They
should regard this as a “quick and dirty” introduction for those who are unfamiliar (or a bit hazy) with linear
algebra.
![](https://csdnimg.cn/release/download_crawler_static/85126626/bg2.jpg)
2 | A Brief Review of Linear Algebra
Instead of solving x + y = 7, 2x + 4y = 22, let us go up one level of abstraction and
consider
ax + by =u (1)
cx +dy =v (2)
Subtracting b times (2) from d times (1), we obtain
(da −bc)x =du − bv (3)
and thus
x =
du − bv
ad − bc
=
1
ad − bc
(d , −b)
u
v
(4)
Note that the scalar product
∗
of a row vector with a column vector naturally appears.
Given a row vector
P
T
= (p, q) and a column vector
Q =
r
s
, their scalar product is
defined to be
P
T
.
Q = (p, q)
r
s
= pr +qs. (The superscript T on
P will be explained
in due time.)
Similarly, subtracting a times (2) from c times (1), we obtain
(cb −ad)y = cu − av (5)
and thus
y =−
cu − av
ad − bc
=
1
ad − bc
(−c, a)
u
v
(6)
(With a =1, b = 1, c = 2, d = 4, u = 7, v =22, we have x =3, y = 4, but this is all child’s
play for the reader, of course.)
Matrix appears
Packaging (4) and (6) together naturally leads us to the notion of a matrix:
†
x
y
=
1
ad − bc
du − bv
−cu + av
=
1
ad − bc
d −b
−ca
u
v
(7)
The second equality indicates how the action of a 2-by-2 matrix on a 2-entry column
vector is defined. A 2-by-2 matrix acting on a 2-entry column vector produces a 2-entry
column vector as follows. Its first entry is given by the scalar product of the first row of the
matrix, regarded as a 2-entry row vector, with the column vector, while its second entry is
given by the scalar product of the second row of the matrix, regarded as a 2-entry row vector,
with the column vector. I presume that most readers of this review are already familiar with
∗
Also called a dot product.
†
“Matrix” comes from the Latin word for womb, which in turn is derived from the word “mater.” The term
was introduced by J. J. Sylvester.
![](https://csdnimg.cn/release/download_crawler_static/85126626/bg3.jpg)
A Brief Review of Linear Algebra | 3
how a matrix acts on a column vector. As another example,
ab
cd
x
y
=
ax + by
cx +dy
(8)
At this point, we realize that we could write (1) and (2) as
ab
cd
x
y
=
u
v
(9)
Define the matrix M =
ab
cd
and write x =
x
y
and u =
u
v
. Then we can express
(9) as
M x =u (10)
Thus, given the matrix M and the vector u, our problem is to find a vector x such that M
acting on it would produce u.
Turning a problem around
As is often the case in mathematics and physics, turning a problem around
1
and looking
at it in a novel way could open up a rich vista. Here, as some readers may know, it is fruitful
to turn (10) around into u = M x and to look at it as a linear transformation of the vector x
by the matrix M into the vector u, conceptualized as M : x →u, rather than as an equation
to solve for x in terms of a given u.
Once we have the notion of a matrix transforming a vector into another vector, we could
ask what happens if another matrix N comes along and transforms u into another vector,
call it p:
p = N u = NMx = P x (11)
The last equality defines the matrix P . At this point, we may become more interested in
how two matrices N and M could be multiplied together to form another matrix P = NM,
and “dump”
2
the vectors p, u, and x altogether, at least for a while.
The multiplication of matrices provides one of the central themes of group theory. We
will see presently that (11) tells us how the product NM is to be determined, but first we
need to introduce indices.
Appearance of indices and rectangular matrices
If we want to generalize this discussion on 2-by-2 matrices to n-by-n matrices, we risk
running out of letters. So, we are compelled to use that marvelous invention known as the
index.
Write M =
M
11
M
12
M
21
M
22
. Here we have adopted the totally standard convention of denot-
ing by M
ij
the entry in the ith row and jth column of M. The reader seeing this for the
![](https://csdnimg.cn/release/download_crawler_static/85126626/bg4.jpg)
4 | A Brief Review of Linear Algebra
first time should make sure that he or she understands the convention about rows and
columns by writing down M
31
and M
23
in the following 3-by-3 matrix:
M =
⎛
⎜
⎜
⎝
abc
def
gh i
⎞
⎟
⎟
⎠
(12)
The answer is given in an endnote.
3
Starting with the chicken and rabbit problem, we were led to square matrices. But we
could just as well define m-by-n rectangular matrices with m rows and n columns. Indeed, a
column vector could be regarded as a rectangular matrix with m rows and only one column.
A row vector is a rectangular matrix, with one row and n columns.
Rectangular
∗
matrices could be multiplied together only if they “match”; thus an m-by-
n rectangular matrix can be multiplied from the right by an n-by-p rectangular matrix to
form an m-by-p rectangular matrix.
Writing x =
x
1
x
2
and u =
u
1
u
2
(which amounts to regarding a vector as a rectangular
matrix with two rows but only one column), we could restate (10) (or in other words, (1) and
(2), the equations we started out with) as u
i
= M
i1
x
1
+ M
i2
x
2
=
2
j=1
M
ij
x
j
, for i =1, 2.
Multiplying matrices together and the Kronecker delta
Now the generalization to n-dimensional vectors and n-by-n matrices is immediate. We
simply allow the indices i and j to run over 1,
...
, n and extend the upper range in the
summation symbol to n:
u
i
=
n
j=1
M
ij
x
j
(13)
The rule for multiplying matrices then follows from (11):
p
i
=
n
j=1
N
ij
u
j
=
n
j=1
n
k=1
N
ij
M
jk
x
k
=
n
k=1
P
ik
x
k
(14)
Hence P = NM means
P
ik
=
n
j=1
N
ij
M
jk
(multiplication rule) (15)
We now define the identity matrix I by I
ij
= δ
ij
, with the Kronecker delta symbol δ
ij
defined by
δ
ij
=
1ifi = j
0ifi = j
(16)
∗
We will seldom encounter rectangular matrices other than vectors; one occasion would occur in the proof
of Schur’s lemma in chapter II.2.
![](https://csdnimg.cn/release/download_crawler_static/85126626/bg5.jpg)
A Brief Review of Linear Algebra | 5
In other words, I is a matrix whose off-diagonal elements all vanish and whose diagonal
elements are all equal to 1. In particular, for n =2, I =
10
01
.
It follows from (15) that (I M)
ik
=
n
j=1
δ
ij
M
jk
=M
ik
. Similarly, (MI )
ik
=M
ik
. In other
words, IM = MI =M.
If the reader feels a bit shaky about matrix multiplication, now is the time to practice with
a few numerical examples.
4
Please also do exercises 1–5, in which the notion of elementary
matrices is introduced; we will need the results later.
I also mention here another way of looking at matrix multiplication that will be useful
later. Regard the n columns in M as n different column vectors
ψ
(k)
, k = 1,
...
, n, where
by definition the j th component of
ψ
(k)
is equal to M
jk
. (The parenthesis is a bit pedantic:
it emphasizes that k labels the different column vectors.) Thus, schematically,
M =
ψ
(1)
,
...
,
ψ
(k)
,
...
,
ψ
(n)
(17)
For example, for the 3-by-3 matrix in (12),
ψ
(1)
=
a
d
g
,
ψ
(2)
=
b
e
h
, and
ψ
(3)
=
c
f
i
.
Similarly, regard the n columns in P as n different column vectors
φ
(k)
, k = 1,
...
, n,
where by definition the jth component of
φ
(k)
is equal to P
jk
. Looked at this way, (15) is
telling us that
φ
(k)
is obtained by acting with the matrix N on
ψ
(k)
:
P =
φ
(1)
,
...
,
φ
(n)
= NM =
N
ψ
(1)
,
...
, N
ψ
(n)
(18)
Einstein’s repeated index summation
Let us now observe that whenever there is a summation symbol, the index being summed
over is repeated. For example, in (15) the summation symbol instructs us to sum over the
index j , and indeed the index j appears twice in N
ij
M
jk
, in contrast to the indices i and
k, which appear only once each and are sometimes called free indices.
Notice that the free indices also appear in the left hand side of (15), namely in P
ik
. This
is of course as it should be: the indices on the two sides of the equation must match. In
contrast, the index j , which has been summed over, must not appear in the left hand side.
As you can see (and as could Einstein), the summation symbol is redundant in many
expressions and may be omitted if we agree that any index that is repeated, such as j in this
example, is understood to be summed over. In the physics literature, Einstein was among
the first to popularize this repeated index summation, which many physicists regard as
one of Einstein’s important contributions.
5
We will adopt this convention and hence write
(15) simply as P
ik
= N
ij
M
jk
.
The index being summed over (namely, j in this expression) is sometimes called the
dummy index, in contrast to the free indices i and k, which take on fixed values we are
free to assign. Here is a self-evident truth seemingly hardly worth mentioning, but yet it
sometimes confuses some abecedarians: it does not matter what we call the dummy index.
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