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Unformatted text preview: 3.4. THE DUAL OF A VECTOR SPACE AND MATRICES Thus the j t h column of ŒS T D;B and of ŒS D;C ŒT C;B agree. 181 I For a vector space V over a ﬁeld K , we denote the set of K –linear
maps from K to K by EndK .V /. Since the composition of linear maps is
linear, EndK .V / has a product .S; T / 7! S T . The reader can check that
EndK .V / with the operations of addition and and composition of linear
operators is a ring with identity. To simplify notation, we write ŒT B instead of ŒT B;B for the matrix of a linear transformation T with respect to
a single basis B of V .
Corollary 3.4.12. Let V be a ﬁnite dimensional vector space over K . Let
n denote the dimension of V and let B be an ordered basis of V .
(a) For all S; T 2 EndK .V /, ŒS T B;B D ŒS B ŒT B .
(b) T 7! ŒT B is a ring isomorphism from EndK .V / to Matn .K/. Lemma 3.4.13. Let B D .v1 ; : : : ; vn / and C D .w1 ; : : : ; wn / be two
bases of a vector space V over a ﬁeld K . Denote the dual bases of V by
B D v1 ; : : : ; vn and C D w1 ; : : : ; wn . Let id denote the identity
linear transformation of V .
(a) The matrix ŒidB;C of the identity transformation with respect to
the bases C and B has .i; j / entry hwj ; vi i:
(b) ŒidB;C is invertible with inverse ŒidC;B . Proof. Part (a) is immediate from the deﬁnition of the matrix of a linear
transformation on page 179. For part (b), note that
E D ŒidB D ŒidB;C ŒidC;B :
Let us consider the problem of determining the matrix of a linear transformation T with respect to two different bases of a vector space V . Let
B and B 0 be two ordered bases of V . Then
ŒT B D ŒidB;B 0 ŒT B 0 ŒidB 0 ;B ;
by an application of Proposition 3.4.11. But the “change of basis matrices”
ŒidB;B 0 and ŒidB 0 B are inverses, by Lemma 3.4.13 Writing Q D ŒidB;B 0 ,
ŒT B D QŒT B 0 Q 1 : ...
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