Page 140 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes a j f ( ), j 1,..., .
n
j
n
Every linear functional on F is of this form, for some scalars a , ..., a . That is immediate from the
1 n
definition of linear functional because we define a = f( ) and use the linearity
j j
f (x , ...., x ) = f x j j
1 n
j

= x f ( )
j
j
j
= a x
j j
j

Example 2: Here is an important example of a linear functional. Let n be a positive
integer and F is field. 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 .
The trace function is a linear functional on the matrix space F n × n because

n
tr (cA B ) = (cA ii B ii )
i 1

n n
= c cA ii B ii
i 1 i 1
= c tr A + tr B.

Example 3: 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
L t ( ) p ( )
t
p
then L is a linear functional on V. One usually describes this by saying that, for each t, ‘evaluation
t
at t’ is a linear functional on the space of polynomial functions. Perhaps we should remark that
the fact that the functions are polynomials plays no role in this example. Evaluation at t is a
linear functional on the space of all functions from F into F.

Example 4: This may be the most important linear functional in mathematics. Let [a, b]
be a closed interval on the real line and let C([a, b]) be the space of continuous real-valued
functions on [a, b]. Then

b
L(g) = g ( ) dt
t
a
Theorem 1: Let V be an n-dimensional vector space over the field F, and let W be an m-dimensional
vector space over F. Then the space L(V, W) is finite-dimensional and has dimension mn.

Proof: Let
, ,... n and , ,...
1 2 1 2 n

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