Page 148 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Self Assessment

5
1. Let W be the subspace of R which is spanned by the vectors
2 , 3 3
1 1 2 3 2 2 3 4 5
4 6 4
3 1 2 3 4 5
Find a basis for W*.
2. Let W be the subspace spanned by R , which is spanned by the vectors
5
(1, 2, 0, 3, 0), (1, 2, 1, 1, 0)
1 2
(0, 0, 1, 4, 0), (2, 4, 1, 10, 1)
3 4
(0, 0, 0, 0, 1)
5
How does one describe W*, the annihilator of W.

10.3 Summary

 The concept of linear functional helps us to clarify the discussion of subspaces, linear
equations and co-ordinates.

 In this unit the idea of dual basis for V* is obtained i.e. if B ( 1 , 2 ,... n ) be the basis of
V then there is a unique dual basis * ( , ... )f 1 f n for V*.

 The concept of linear functional is important in the study of finite-dimensional spaces
because it helps to organize and clarify the discussion of subspaces, linear equations, and
coordinates.
 Let V be the space of all polynomial functions from the field F into itself. Let t be an
element of F. If we define

p
L t ( ) p ( )
t
then L is a linear functional on V. One usually describes this by saying that, for each t,
t
‘evaluation at t’ is a linear functional on the space of polynomial functions.
10.4 Keywords

Dual Basis: In particular, if f is the zero functional f( ) = 0 for each j and hence the scalars c are
j j
all 0. Now f 1 ,...f are n linearly independent functionals, and since we know that V* has
n
dimension n, it must be that * { ,...,f 1 f n } is a basis for V*. This basis is called the dual basis
of .
Linear Functional: If V is a vector space over the field F, a linear transformation f from V into the
scalar field F is also called a linear functional on V.

Trace: If A is an n × n matrix with entries in F, the trace of A is the scalar tr A A 11 A 22 ... A nn .

10.5 Review Questions

1. In R 3 , (1, 0, 1) (0, 1, – 2), (–1, 1, 0)
1 2 3
If f is a linear functional on R such that
3

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