Page 132 - Linear Algebra Done Right

Linear Functionals and Adjoints
119
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T (y 1 ,y 2 ) = (2y 2 ,y 1 , 3y 1 ).
turned out to be not just a func-
Note that in the example above, T
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3
2
tion from R to R , but a linear map. That is true in general. Specif- Adjoints play a crucial
role in the important
ically, if T ∈L(V, W), then T ∗ ∈L(W, V). To prove this, suppose results in the next
T ∈L(V, W). Let’s begin by checking additivity. Fix w 1 ,w 2 ∈ W. chapter.
Then
Tv, w 1 + w 2 = Tv, w 1 + Tv, w 2
∗ ∗
= v, T w 1 + v, T w 2
= v, T w 1 + T w 2 ,
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which shows that T w 1 +T w 2 plays the role required of T (w 1 +w 2 ).
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Because only one vector can behave that way, we must have
T w 1 + T w 2 = T (w 1 + w 2 ).
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Now let’s check the homogeneity of T .If a ∈ F, then
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Tv, aw = ¯ a Tv, w
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= ¯ a v, T w
= v, aT w ,
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which shows that aT w plays the role required of T (aw). Because
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only one vector can behave that way, we must have
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aT w = T (aw).
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Thus T ∗ is a linear map, as claimed.
You should verify that the function T T ∗ has the following prop-
erties:
additivity
(S + T) = S + T ∗ for all S, T ∈L(V, W);
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conjugate homogeneity
(aT) = ¯ aT ∗ for all a ∈ F and T ∈L(V, W);
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adjoint of adjoint
(T ) = T for all T ∈L(V, W);
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identity
I = I, where I is the identity operator on V;
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