transformation A and sometimes denoted by Im A. We have a so-called “dimension
theorem” expressed as follows:
Theorem 11.3: Dimension Theorem Let V
n be a linear vector space of dimension
n. Also let A be an endomorphism: V
n
! V
n . Then we have
dim V
n
¼ dimA V
n
ð Þ þ dim Ker A:
ð11:45Þ
The number of dimA(V
n ) is said to be a rank of the linear transformation A. That is,
we write
dimA V
n
ð Þ ¼ rank A:
Also the number of dim Ker A is said to be a nullity of the linear transformation A.
That is, we have
dim Ker A ¼ nullity A:
Thus, (11.45) can be written succinctly as
dim V
n
¼ rank A þ nullity A:
Proof Let e 1 , e 2 , Á Á Á, and e n be basis vectors of V
n . First, assume
A e 1
ð Þ ¼ A e 2
ð Þ ¼ Á Á Á ¼ A e n
ð Þ ¼ 0:
This implies that nullity A ¼ n. Then, A ∑
n
i¼1 x i e i
À
Á ¼ ∑
n
i¼1 x i A e i
ð Þ ¼ 0. Since x i is
arbitrarily chosen, the expression means that A(x) ¼ 0 for
8
x 2 V
n . This implies
A ¼ 0. That is, rank A ¼ 0. Thus, (11.45) certainly holds.
To proceed with proof of the theorem, we think of a linear combination ∑
n
i¼1 c i e i .
Next, assume that Ker A ¼ Span{e 1 , e 2 , Á Á Á, e ν } (ν < n); dim Ker A ¼ ν. After A is
operated on the above linear combination, we are left with ∑
n
i¼νþ1 c i A e i
ð Þ. We put
∑
n
i¼1 c i A e i
ð Þ ¼ ∑
n
i¼νþ1 c i A e i
ð Þ ¼ 0:
ð11:46Þ
Suppose that the (n À ν) vectors A(e i ) (ν + 1 i n) are linearly dependent. Then
without loss of generality we can assume c ν + 1 6 ¼ 0. Dividing (11.46) by c ν + 1 we
obtain
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11 Vectors and Their Transformation