Page 134 - Linear Algebra Done Right

Linear Functionals and Adjoints
we are dealing with orthonormal bases—with respect to nonorthonor-
mal bases, the matrix of T
∗
transpose of the matrix of T. does not necessarily equal the conjugate 121
6.47 Proposition: Suppose T ∈L(V, W).If (e 1 ,...,e n ) is an or- The adjoint of a linear
thonormal basis of V and (f 1 ,...,f m ) is an orthonormal basis of W, map does not depend
then on a choice of basis.

M T ,(f 1 ,...,f m ), (e 1 ,...,e n ) This explains why we
∗
will emphasize adjoints
is the conjugate transpose of
of linear maps instead
of conjugate
M T, (e 1 ,...,e n ), (f 1 ,...,f m ) .
transposes of matrices.
Proof: Suppose that (e 1 ,...,e n ) is an orthonormal basis of V and
(f 1 ,...,f m ) is an orthonormal basis of W. We write M(T) instead of the

longer expression M T, (e 1 ,...,e n ), (f 1 ,...,f m ) ; we also write M(T )
∗

instead of M T ,(f 1 ,...,f m ), (e 1 ,...,e n ) .
∗
Recall that we obtain the k th column of M(T) by writing Te k as a lin-
ear combination of the f j ’s; the scalars used in this linear combination
then become the k th column of M(T). Because (f 1 ,...,f m ) is an ortho-
normal basis of W, we know how to write Te k as a linear combination
of the f j ’s (see 6.17):
Te k = Te k ,f 1 f 1 +· · ·+PTe k ,f m f m .

Thus the entry in row j, column k,of M(T) is Te k ,f j . Replacing T
with T ∗ and interchanging the roles played by the e’s and f ’s, we see
that the entry in row j, column k,of M(T ) is T f k ,e j , which equals
∗
∗
f k ,Te j , which equals Te j ,f k , which equals the complex conjugate
of the entry in row k, column j,of M(T). In other words, M(T ) equals
∗
the conjugate transpose of M(T).