Matrix of linear maps. Matrix-vector product

Transcription

1 Matrix of linear maps. Matrix-vector product Aim lecture: Introduce matrices of linear maps as a way of understanding more complicated linear maps. Consider direct sums of F-spaces V = m j=1 V j, W = n i=1 W i. Notn Write (L(V j, W i )) ij for the direct sum m j=1 n i=1 L(V j, W i ) except the elements are not written as column vectors but in an n m-matrix so has form T = (T ij ) ij where the entries T ij : V j W i are linear & i = 1,..., n, j = 1,..., m. Given v = (v 1,..., v m ) T V and T = (T ij ) ij (L(V j, W i )) ij we can define the matrix-vector product T v which is the element w of W whose i-th entry is w i = m T ij v j. j=1 Daniel Chan (UNSW) Lecture 11: Matrices of linear maps Semester / 9

2 Evaluation map & examples of matrix-vector product It is an easy exercise to prove Prop-Defn For a set X & x X, the evaluation at x map ev x : Fun(X, F) F defined by ev x (f ) = f (x) is linear. E.g.1 We can put the lin maps d dx : R[x] R[x], ev 0 : R[x] R, 2 id : R R into a matrix to define ( d ) ( ) dx 0 f (x) = ev 0 2 id β for f (x) R[x], β F. E.g.2 We may re-write the initial value problem as the single matrix eqn d 2 y 4y = sin x, y(0) = 3 dy 2 Daniel Chan (UNSW) Lecture 11: Matrices of linear maps Semester / 9

3 L(F, V ) Prop Let V = F-space. 1 For any v V, the map T v : F V defined by T v β = βv is linear. 2 Every F-lin map T : F V can be written uniquely in the form T v for some v V as above. 3 In fancier language, there is a natural isomorphism V L(F, V ) : v T v. Proof. For 1) just note for γ, β, β F we have T v (γβ + β ) = (γβ + β )v = γβv + β v = γt v β + T v β. For 2), given lin T : F V we let v = T (1) and note T = T v since for any γ F we have T (γ) = γt (1) = γv = T v γ. No other choice of vector v works so it is unique. You can check as ex that the bijection V L(F, V ) : v T v is indeed linear so 3) follows. Upshot: Hence we will often identify L(F, V ) with V & in particular, L(F, F) with F Daniel Chan (UNSW) Lecture 11: Matrices of linear maps Semester / 9

4 Connection with old matrix product Consider now the situation V = F n = n j=1 F, W = Fm = m i=1 F. Then a matrix A M mn (F) in the old sense is also a matrix in (L(F, F)) ij in the new sense. Moreover, for v V the two defns (old & new) of Av define the same element of W. Daniel Chan (UNSW) Lecture 11: Matrices of linear maps Semester / 9

5 Linearity of matrix-vector product Consider direct sums of F-spaces V = m j=1 V j, W = n i=1 W i. Below we write (v j ) j as an abbreviated form for (v 1,..., v m ) T. Prop Consider a matrix of linear maps T = (T ij ) ij (L(V j, W i )) ij. Then (abusing notn) the associated map T : V W : v T v is linear. Proof. For v = (v j ) j, v = (v j ) j V, β F. Then T (βv + v ) = T (βv j + v j) j = ( j T ij (βv j + v j)) i = ( j (βt ij v j + T ij v j)) i = (β j T ij v j + j T ij v j) i This completes the proof. = βt v + T v. Daniel Chan (UNSW) Lecture 11: Matrices of linear maps Semester / 9

6 Identifying matrices with associated linear maps Rem We will often confuse a matrix as above with the associated linear map induced by the matrix product. What permits us to do this is the following fact. Fact Distinct matrices induce distinct linear maps. Proof. Follows by generalising the example below & will be proved in full generality in proof of matrix reprn thm. E.g. Consider the map ( ) f (x) T : F[x] F F F : β ( ) f (1) + 2β. f (0) 3β 1 Show that T is linear by finding a 2 2-matrix of linear maps inducing it. 2 Explain why the 2 2-matrix found above is uniquely determined by T. Daniel Chan (UNSW) Lecture 11: Matrices of linear maps Semester / 9

7 Direct sums of linear maps Prop-Defn For i = 1,..., m, let T i : V i W i be linear maps. We define the map m i=1 T i = T 1... T m : V 1... V m W 1... W m by ( m i=1 T i)(v 1,..., v m ) T = (T 1 v 1,..., T m v m ) T. It is a linear map called the direct sum of the T i. Proof. Linearity follows from the fact that m i=1 T i corresponds to the diagonal matrix (T ij ) ij (L(V j, W i )) ij with T ii = T i & T ij = 0 if i j. This is easily seen from any E.g. T 1 = d dx : C (R) C (R), T 2 = (2 3) : R 2 R. Daniel Chan (UNSW) Lecture 11: Matrices of linear maps Semester / 9

8 Example of matrices of matrices Matrices define linear maps so can be the entries of a matrix of linear maps too. E.g. Consider the 4 matrices ( ) ( ) A 11 =, A =, A = ( 9 10 ), A 22 = ( ) Then identifying F 4 = F 2 F 2, F 3 = F 2 F 1 we find ( ) x 1 A11 A 12 x 2 A 21 A 22 x 3 x 4 which is the linear map associated to the 3 4-matrix obtained by removing internal brackets. Daniel Chan (UNSW) Lecture 11: Matrices of linear maps Semester / 9

9 General result concerning matrices of matrices We can generalise this result as follows. Let m = m m M, n = n n N. We identify F m = F m1... F m M, F n = F n1... F n N. Prop For i = 1,..., M, j = 1,..., N let A ij M mi n j so that we may form the M N-matrix (A ij ) ij of matrices. The associated linear map from F n F m is induced by the m n-matrix obtained from (A ij ) ij by removing internal brackets. Proof omitted. The proof is notationally messy but easy and follows as in the previous example. Motto A matrix of matrices is really just one big fat matrix (of scalars)! Daniel Chan (UNSW) Lecture 11: Matrices of linear maps Semester / 9