Page 155 - DMTH502_LINEAR_ALGEBRA

Unit 11: The Double Dual

(b) Show that the dual space W* of W can be ‘naturally’ identified with the linear Notes
functionals

f x 1 ,...,x n c x 1 ... c x n
n
1
n
on F which satisfy c 1 ... c n 0.
2. Use Theorem 4 to prove the following. If W is a subspace of a finite-dimensional vector
space V and if g ,...,g is any basis for W°, then
1 r
r
W N .
i 1 g
11.2 The Transpose of a Linear Transformation

Suppose that we have two vector spaces over the field F, V and W, and a linear transformation T
from V into W. Then T induces a linear transformation from W* into V*, as follows. Suppose g is
a linear functional on W, and let
f g T ...(11)

for each in V. Then (11) defines a function f from V into F, namely the composition of T, a
function from V into W, with g, a function from W into F. Since both T and g are linear, Theorem
5 of unit 7 tells us that f is also linear, i.e., f is a linear functional on V. Thus T provides us with a
t
t
rule T which associates with each linear functional g on W a linear functional f = T g on V, defined
by (11). Note also that T is actually a linear transformation from W* into V*; for, if g and g are
t
1 2
in W* and c is a scalar
t
T cg 1 g 2 cg 1 g 2 T
cg T g T
2
1
t
t
c T g 1 T g 2 ...(12)
t
t
t
.
so that T cg g cT g T g 2 Let us summarize.
1 2 1
Theorem 5: Let V and W be vector spaces over the field F. For each linear transformation T from
t
V into W, there is a unique linear transformation T from W* into V* such that
t
T g g T ...(13)
for every g in W* and in V.
t
We shall call T the transpose of T. This transformation T is often called the adjoint of T; however,
t
we shall not use this terminology.
Theorem 6: Let V and W be vector spaces over the field F, and let T be a linear transformation
from V into W. The null space of T is the annihilator of the range of T. If V and W are finite-
t
dimensional, then
t
(i) rank T = rank (T)
t
(ii) the range of T is the annihilator of the null space T. ...(14)

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