Page 141 - DMTH502_LINEAR_ALGEBRA

Unit 10: Linear Functionals

be ordered basis for V and W, respectively. For each pair of integers (p, q) with 1 p m and Notes
p, q
1 q n, we define a linear transformation E from V into W by
0 if i q
E , p q ( i ) =
p if i q

=
iq b
According to the theorem 1 of unit 7, there is a unique linear transformation from V into W
satisfying these conditions. The claim is that the mn transformations E from a basis for L(V, W).
p, q
Let T be a linear transformation from V into W. For each j, i j n, let A , A , ... A be the
1j 2j mj
co-ordinates of the vector T in the ordered basis , i.e.,
j
m
T = A ...(1)
j pj p
p 1
we wish to show that

m n
T = A E , p q ...(2)
pq
p 1 q 1
Let U be the linear transformation in the right hand member of (2). Then for each j

U = A E , p q ( )
j pq j
p q

= A pq jq p
p q

m
= A pj p
p 1
= T
j
p, q
and consequently U = T. Now (2) shows that the E span L(V, W); we must prove that they are
independent. But this is clear from what we did above; for, if the transformation

U = A E , p q
pq
p q
is the zero transformation, then U = 0 for each j, so
j
m
Ap j j = 0
p 1

and the independence of the implies that A = 0 for every p and j.
p pj
If V is finite-dimensional vector space, the collection of linear functionals of V forms a vector
space in a natural way. It is the space L(V, F). We denote this space by V*. From the above
theorem we know the following about the space V* that
dim V* = dim V. ...(3)

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