CHAPTER II DIMENSION In the present chapter we investigate ...

Chapter III will be devoted to the basic skill of handling the summation symbol. You may
return to the second proof after studying that section.
Again let E and F be two bases in V , where E consists of vectors e1, e2, . . . , em and
F consists of f1, f2, . . . , fn. Now each vector in E can be written as a linear combination
of vectors in F and vice versa, say
ej = n
For each ℓ with 1 ≤ ℓ ≤ n, we have
k= 1 pjkfk and fk = m
j= 1 qkjej.
fℓ =
j qℓjej =
j qℓj
k pjkfk =
j
=
k j qℓjpjkfk =
k j qℓjpjk
fk.
k qℓjpjkfk
Since f1, f2, . . . , fn form a basis, the coefficient of fk must be the same for each k, equal
to 1 for k = ℓ and zero otherwise. Thus we have
j qℓjpjk
1,
= δℓk ≡
0,
if k = ℓ;
otherwise.
(2.2.1)
(Here, δℓk given above is the so-called Kronecker’s delta.) We can rewrite (2.2.1) in
matrix form as QP = In, where In is the n × n identity matrix and
⎡
⎤ ⎡
⎤
p11 · · · · · · p1m
q11 · · · · · · q1n
⎢
P =
.
⎣ .
. ⎥ ⎢ .
. ⎥
.
. ⎦ , Q = ⎣ .
. ⎦ .
pn1 · · · · · · pnm
qm1 · · · · · · qmn
By reversing the roles of p’s and q’s, we know that (2.2.1) holds if p and q are switched,
giving us P Q = Im.
♠ Aside: Merely P Q = In alone is not enough to guarantee that Q is the inverse of P
(unless you also know that both P and Q are square matrices) One could have AB = In
without the invertibility of A and B. For example, if
A =
⎡ ⎤
1 0
1 0 0
, B = ⎣ 0 1 ⎦ ,
0 1 0
9 9
then AB = I2, but A, B are not invertible because they are not square matrices. ♠
Consider the sum S of all pkjqjk (= qjkpkj), where k runs from 1 to n and j runs
from 1 to m. We can perform the addition in two different ways: one way is letting k run
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