Page 156 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Proof: If g is in W*, then by definition
t
T g g T

t
for each in V. The statement that g is in the null space of T means that g T 0 for every
t
in V. Thus the null space of T is precisely the annihilator of the range of T.
Suppose that V and W are finite-dimensional, say dim V = n and dim W = m. For (i): Let r be the
rank of T, i.e., the dimension of the range of T. By Theorem 3 of unit 10, the annihilator of the
.
range of T then has dimension m r By the first statement of this theorem, the nullity of T t
t
.
must be m r But then since T is a linear transformation on an m-dimensional space, the rank
t
of T is m m r , r and so T and T have the same rank. For (ii): Let N be the null space of T.
t
t
t
Every functional in the range of T is in the annihilator of N; for suppose f = T g for some g in W*;
then, if is in N
t
f T g g T g 0 0.
t
0
Now the range of T is a subspace of the space N , and
0
dim N = n –dim N = rank(T) = rank (T ) t ...(15)
so that the range of T must be exactly N .
0
t
Theorem 7: Let V and W be finite-dimensional vector spaces over the field F. Let  be an ordered
basis for V with dual basis *, and let ' be an ordered basis for W with dual basis '*. Let T be
a linear transformation from V into W; let A be the matrix of T relative to , '  and let B be the
matrix of T relative to '*, *.  Then B A .
t
ij ji
Proof: Let


 = ,..., , '= ,..., ,
1 n 1 m
 *= f 1 ,...,f n , '*= g 1 ,...,g m .

By definition,

m
T A i j 1,...,n
j ij i
i 1
n ...(16)
t
T g j B f j j 1,...,m
ij i
i 1
On the other hand,

t
T g g T
j i j i
m
g j A ki k
k 1

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