Page 130 - Linear Algebra Done Right

Linear Functionals and Adjoints
Linear Functionals and Adjoints
A linear functional on V is a linear map from V to the scalars F. 117
3
For example, the function ϕ: F → F defined by
6.43 ϕ(z 1 ,z 2 ,z 3 ) = 2z 1 − 5z 2 + z 3
3
is a linear functional on F . As another example, consider the inner-
product space P 6 (R) (here the inner product is multiplication followed
by integration on [0, 1]; see 6.2). The function ϕ: P 6 (R) → R defined
by
1
6.44 ϕ(p) = p(x)(cos x) dx
0
is a linear functional on P 6 (R).
If v ∈ V, then the map that sends u to u, v is a linear functional
on V. The next result shows that every linear functional on V is of this
form. To illustrate this theorem, note that for the linear functional ϕ
3
defined by 6.43, we can take v = (2, −5, 1) ∈ F . The linear functional
ϕ defined by 6.44 better illustrates the power of the theorem below be-
cause for this linear functional, there is no obvious candidate for v (the
function cos x is not eligible because it is not an element of P 6 (R)).

6.45 Theorem: Suppose ϕ is a linear functional on V. Then there is
a unique vector v ∈ V such that

ϕ(u) = u, v
for every u ∈ V.

Proof: First we show that there exists a vector v ∈ V such that
ϕ(u) = u, v for every u ∈ V. Let (e 1 ,...,e n ) be an orthonormal
basis of V. Then

ϕ(u) = ϕ( u, e 1 e 1 + ··· + u, e n e n )
= u, e 1 ϕ(e 1 ) + ··· + u, e n ϕ(e n )

= u, ϕ(e 1 )e 1 + ··· + ϕ(e n )e n
for every u ∈ V, where the first equality comes from 6.17. Thus setting
v = ϕ(e 1 )e 1 +· · ·+ ϕ(e n )e n , we have ϕ(u) = u, v for every u ∈ V,
as desired.