N x i 1 a i w i it is clear that with this definition

This preview shows page 28 - 31 out of 64 pages.

= n X i =1 a i w i . It is clear that with this definition T ( v i ) = w i , as required. It remains to check that T is linear and that T is the only linear transformation such that maps v i to w i for all i = 1 , 2 , . . . , n . We first check that T is linear. Let v = n X i =1 a i v i , u = n X i =1 b i v i . 28
Then T ( αv + βu ) = T n X i =1 ( αa i + βb i ) v i ! = n X i =1 ( αa i + βb i ) w i = αT ( v ) + βT ( u ) . Now suppose there is another linear transformation U : V W such that U ( v i ) = w i for all i = 1 , 2 , . . . , n . Writing v = n X i =1 a i v i again, we have (because U is linear) U ( v ) = n X i =1 a i U ( v i ) = n X i =1 a i w i = T ( v ) , and so T = U . Corollary 17. Let V and W be vector spaces , and suppose V is finite dimensional with a basis { v 1 , . . . , v n } . If T, U : V W are two linear transformations such that T ( v i ) = U ( v i ) for all i = 1 , 2 , . . . , n then T = U . Example: We consider two bases in R 3 : B = { e 1 = (1 , 0 , 0) , e 2 = (0 , 1 , 0) , e 3 = (0 , 0 , 1) } ˜ B = { v 1 = (1 , 0 , - 1) , v 2 = (1 , 0 , 1) , v 3 = (0 , 1 , 0) } . We wish to find a linear transformation such that T ( e 1 ) = v 1 , T ( e 2 ) = v 2 , T ( e 3 ) = v 3 . By the theorem above, we must have T a b c = T ( a 1 e 1 + a 2 e 2 + a 3 e 3 ) = a 1 v 1 + a 2 v 2 + a 3 v 3 = a 1 + a 2 a 3 - a 1 + a 2 . In fact, if you multiply the matrices out, you’ll find that T a 1 1 2 a 3 = 1 1 0 0 0 1 - 1 1 0 a 1 a 2 a 3 . We have already seen this relationship between linear transformations and matrices in the case of linear transformations mapping R n to R m . We will see in the next section that, after choosing bases for the domain and target of a linear transforma- tion, one obtains the same sort of matrix representation of a linear mapping. However, before we do that, we need several more terms. Definition 9. Let V, W be vector spaces. A linear transformation T : V W which is both one-to-one and onto is called a (linear) isomorphism. Two vector spaces V, W are isomorphic if there exists a (linear) isomorphism T : V W . We write this condition as V W . Isomorphic is a Greek word; it means ”same structure.” 29
Corollary 18. Let V, W be vector spaces and let T : V W be linear. Then the restriction T : V → R ( T ) is an isomorphism if and only if dim(ker( T )) = 0 . This follows from the fact that T is one-to-one if and only if dim(ker( T )) = 0. Corollary 19. Let V, W be finite dimensional vector spaces of the same dimension, and let T : V W be linear. Then T is an isomorphism if and only if T is one- to-one if and only if T is onto. Example: Let V and W be finite dimensional vector spaces, and let dim( V ) 6 = dim( W ). Then V and W cannot be isomorphic. Indeed, suppose that T : V W is an isomorphism. Then, because T would have to be both one-to-one and onto, we must have dim( V ) = dim( R ( T )) + dim(ker( T )) = dim( W ) + 0 , which is impossible. Corollary 20. Let V be a finite dimensional vector space with dim( V ) = n . Then V is isomorphic to R n .

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture