Linear Algebra II Lecture 8

Transcription

1 Linear Algebra II Lecture 8 Xi Chen 1 1 University of Alberta October 10, 2014

2 Outline 1 2

3 Definition Let T 1 : V W and T 2 : V W be linear transformations between two vector spaces V and W over R. Then T 1 + T 2 : V W is the map (T 1 + T 2 )(v) = T 1 (v) + T 2 (v). Let T : V W be a linear transformation between two vector spaces V and W over R and c R. Then ct : V W is the map (ct )(v) = ct (v). Let T 1 : V W and T 2 : U V be linear transformations between vector spaces U, V and W. Then T 1 T 2 is the map (T 1 T 2 )(u) = T 1 (T 2 (u)).

4 Vector Space L(V, W ) Theorem For all linear transformations T 1 : V W and T 2 : V W and c R, T 1 + T 2 and ct 1 are also linear transformations from V to W. Furthermore, [T 1 + T 2 ] B2 B 1 = [T 1 ] B2 B 1 + [T 2 ] B2 B 1 and [ct 1 ] B2 B 1 = c[t 1 ] B2 B 1 where B 1 is a basis for V and B 2 is a basis for W. Let L(V, W ) be the set of all linear transformations from V to W. Then L(V, W ) is itself a vector space over R under the addition and scalar multiplication defined above.

5 Composition of Linear Transformations Theorem Let T 1 : V W and T 2 : U V be linear transformations between vector spaces U, V and W. Then T 1 T 2 is a linear transformation from U W. Furthermore, [T 1 T 2 ] B3 B 1 = [T 1 ] B3 B 2 [T 2 ] B2 B 1 where B 1, B 2, B 3 are bases for U, V, W, respectively. (T 1 T 2 ) T 3 = T 1 (T 2 T 3 ) c(t 1 T 2 ) = (ct 1 ) T 2 = T 1 (ct 2 ) T 1 (T 2 + T 3 ) = T 1 T 2 + T 1 T 3 (T 1 + T 2 ) T 3 = T 1 T 3 + T 2 T 3.

6 Example of Let T 1 (x, y) = (x + y, x y) and T 2 (x, y) = (y, x) be two linear transformations from R 2 R 2. Then (T 1 + T 2 )(x, y) = T 1 (x, y) + T 2 (x, y) = (x + 2y, 2x y) (2T 1 )(x, y) = 2T 1 (x, y) = (2x + 2y, 2x 2y) and (2T 2 )(x, y) = 2T 2 (x, y) = (2y, 2x) T 1 T 2 (x, y) = T 1 (T 2 (x, y)) = T 1 (y, x) = (y + x, y x) T 2 T 1 (x, y) = T 2 (T 1 (x, y)) = T 2 (x + y, x y) = (x y, x + y) Note that T 1 T 2 T 2 T 1!!!

7 Example of Let B be the standard basis. Then [ ] 1 1 [T 1 ] = [T 1 ] B B = and [T ] = [T 2 ] B B = (T 1 + T 2 )(x, y) = (x + 2y, 2x y) [ ] 1 2 [T 1 + T 2 ] = = [T ] + [T 2 ] [ ] (2T 1 )(x, y) = (2x + 2y, 2x 2y) [ ] 2 2 and [2T 1 ] = = 2[T ] (2T 2 )(x, y) = (2y, 2x) [ ] 0 2 [2T 2 ] = = 2[T ]

8 Example of T 1 T 2 (x, y) = (x + y, x + y) [ ] [ ] [ ] [T 1 T 2 ] = = = [T ][T 2 ] T 2 T 1 (x, y) = (x y, x + y) [ ] [ ] [ ] [T 2 T 1 ] = = Note that [T 1 ][T 2 ] [T 2 ][T 1 ] T 1 T 2 T 2 T 1 = [T 2 ][T 1 ]

9 Definition Let T : V W be a linear transformation from V to W. The kernel of T is K (T ) = ker(t ) = {x V : T (x) = 0} V. The range of T is the image of T, i.e., R(T ) = T (V ) = {T (x) : x V } W. Theorem Let T : V W be a linear transformation from V to W. Then K (T ) is a subspace of V and R(T ) is a subspace of W. Let T (x, y) = (x, x) be a linear transformation from R 2 R 2. Then K (T ) = {(x, y) : T (x, y) = (0, 0)} = {(x, y) : x = 0} and R(T ) = {T (x, y)} = {(x, x)} = {(x, y) : x y = 0}.

10 Proof that K (T ) and R(T ) are subspaces K (T ) is a subspace of V. Since T (0) = 0, 0 K (T ). For all v 1, v 2 K (T ), T (v 1 ) = T (v 2 ) = 0 and hence T (v 1 + cv 2 ) = T (v 1 ) + ct (v 2 ) = 0 for all c R. Therefore, v 1 + cv 2 K (T ). R(T ) is a subspace of W. Since T (0) R(T ), 0 R(T ). For all w 1, w 2 R(T ), there exist v 1, v 2 V such that w 1 = T (v 1 ) and w 2 = T (v 2 ). Thus, w 1 + cw 2 = T (v 1 ) + ct (v 2 ) = T (v 1 + cv 2 ) R(T ).

11 Range and Rank If R(T ) is finite-dimensional, then the dimension of R(T ) is called the rank of T, denoted by rank(t ) = dim R(T ) = dim T (V ). Given a basis B = {v 1, v 2,..., v n } of V, then the range of a linear transformation T : V W is Note that R(T ) = T (V ) = Span{T (v 1 ), T (v 2 ),..., T (v n )}. rank(t ) = dim R(T ) = dim Span{T (v 1 ), T (v 2 ),..., T (v n )} n = dim V.

12 Kernel, Range and Rank of T : R n R m Let T : R n R m be a linear transformation given by T (v) = Av for an m n matrix A. Then K (T ) = {v : T (v) = 0} = {v : Av = 0} = Nul(A). R(T ) = Span{T (e 1 ), T (e 2 ),..., T (e n )} = Span{Ae 1, Ae 2,..., Ae n } = Col(A). rank(t ) = dim R(T ) = dim Col(A) = rank(a).