Page 159 - DMTH502_LINEAR_ALGEBRA

Unit 11: The Double Dual

Notes
–1
T = U TU
'  
= U –1 T U
   ...(25)
= U –1 T  U 

and this is precisely the change-of-basis formula which we derived earlier.

Self Assessment

3. Let V be a finite dimensional vector space over the field F and let T be a linear operator on
V. Let C be a scalar and suppose there is a non-zero vector in V such that T c . Prove
that there is a non-zero linear functional F on V such that 'T f cf .
4. For all A, B matrices in F , prove that–
m
(a) A ' ' A
(b) A B ' A B '
'
(c) AB ' B ' '
A
11.3 Summary

 A vector induces a linear functional in V* and the mapping L is an isomorphism
of V and V**.
 If T is the linear transformation from V into W then it also induces a transformation from
W* into V* through its transpose.

 The alternate name of the transpose transformation is word adjoint transformation.

11.4 Keywords

Adjoint: T is the transpose of T. This transformation T is often called the adjoint of T.
t
t
t
Transpose: If A is an m × n matrix over the field F, the transpose of A is n × m matrix A defined
by A t ij A ji .

11.5 Review Questions

1. Let S be a set, F a field and V(S,F) the space of all functions from S into F:

f g x f x g x
cf x cf x .

Let W be any n-dimensional space of V S ,F . Show that there exists points x 1 ,...,x in S
n
and functions f 1 , ,..., f in W such that f x j S ij .
f
n
2
i

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