7. The Elements of Vector Spaces
9.3 Exercises 124 10. Linear Transformations and Matrices 129
1. Letk(x)be afixed polynomial. Define functions
Mk(x) : P(lR)- +P(lR), Lk(x) : P(lR)- +P(lR),
by
Mk(x)(p(xằ =k(x)p(x),
x
Lk(x)(p(xằ = Jp(t)k(t)dt.
o
Ifk(x) =1 for allx ElR,we drop the subscript and write M andL.
(1) Show that Mk(x) and Lk(x) are linear transformations for all polynomialshex). Calculate
Mx2+2xa(X+x4) and Lx+x2(1+x3).
(2) Calculate D . Mx - Mx .D :P(lR)- + P(lR), where D is defined in Example 2 of Section 8.1. That is find aformula for
(Dã Mx - Mx •D)(p(xằ.
(3) Show that L .Mk(x) =Lk(x) : P(lR)- + P(lR).
9.3 Exercises 125 2. Show that T : IR2- + IR2 defined by
T(x, y) =(ax+by, cx+dy),
a,b,c,d fixed real numbers, is a linear transformation and that Tis an isomorphism if and only ifad ~ bc.
3. Show that T : IR3- + IR3defined by
T(x, y, z)=(alx+a2Y+a3Z, blx+b2y +b3z, CIX+C2Y+C3 Z) is a linear transformation. Any linear transformation from IR3to IR3 takes this form, whereai,bj ,Ckare scalars.
4. Let T : IR2- + IR2 be defined byT(x, y)=(x - y, 2x +y) and let 8 : IR2- +IR2 be defined by8(x, y)=(y - 2x, x+y).
(1) Find T . 8(1,0), (T+8)(1, 0), 8 . T(1, 0), (2T - 8)(1,0),82(1,0), and T2(1, 0).
(2) Find T .8(x, y)and 8 .T(x,y).
(3) What are the vectors(x, y)satisfyingT(x, y)= (1, O)?
5. Let T : IR2- + IR2 be defined byT(x, y)=(x, x). What are the kernel of T and the image of T?
6. Let T : IR3- +IR3be defined byT(x, y, z)=(x+y, y +Z, Z+x). Is T an isomorphism? Find T(1, 0, 0),no,1,0) and T(O, 0,1).
7. Let T : IR3- + IR3be a linear transformation satisfying the condition T(x, y, z)= (0, 0, 0) whenever 2x - y +Z= O.
Let T(O, 0, 1)=(1,2,3). Find T(1, 0, 0) and T(O, 1,0). What is dim(lm(T))?
8. Let T : IR3- + IR3be the linear transformation given by the formula T(x, y, z)= (y +Z, x+Z, x+y).
Show that T is an isomorphism and find an inverse for T.
9. Let0/ be a finite-dimensional vector space over IR, and E I , ... , En a basis for0/. Define for each i,1~i ~ n,a function
E;: 0 / - +IR by the formula
1~i ~n, where
A =aIEl+ ... +anEn .
(1) Show that Ei, ... , E~ are linear transformations.
(2) Find a basis for ker(Ei).
(3) LetS :'V~ L(V, m)be the linear extension of the map i=l, ...,n.
Show thatSis an isomorphism.
(4) Compute dimRL('V,m).
10. Let'Vbe a vector space and define 'V* =L( 'V,m). Define a linear transfonnation
by
B :'V~ 'V** (=L('V*,m)=L(L('V,m), mằ
B(A)(f)= f(A).
(1) Show that B is a linear transfonnation.
(2) Show that ker(B)={O}.
(3) Show that B is an isomorphism if'Vis finite-dimensional.
(4) Show by an example that Im(B):f 'V** when 'Vis not finite- dimensional.
11. LetE ={El , ... , En} be a basis for the finite-dimensional vector space 'V. Show that the linear extension construction gives an isomor- phism
L :Fun(E)~ 'V by
12. LetE = {El , ... , En} be a basis for the finite-dimensional vector space'l!. Show that the map (remember,'V*=L( 'V,m)and the remarks following Theorem 8.5.6)
C :'V*~Fun(E) defined by
C(T)(Ej ) =T(Ej ),
is a vectorspace isomorphism.
13. Define a function
i =1, ... , n,
by
S(p(xằ =p(x+1).
Detennine whether S is a linear transfonnation.
14. Let 'V and 'W be finite-dimensional vector spaces. SupposeE = {El, ... , En} is a basis for 'V and {Fl , ... , Fm} a basis for 'W. For
9.3 Exercises 127 each ordered pair(i,j)show that the function Mi,) :'11---+ 'W, defined by the formula
is a linear transformation.
(1) Show that{Mi,)11:S;i :s;n,l:S;j:S;} is a basis forL(V,W).
(2) Show that dim(L('1I,'Wằ =nm.
15. Let'11be a vector space and suppose that5,TE'1I*= L('11,JR). Define L :'11---+ JR2
by
L(A)=(S(A), T(Aằ.
(1) Show that L is a linear transformation.
(2) Show that ker(L)=ker(S)n ker(T).
16. Let'11be a vector space and define (see Theorem 8.5.6)
by
Ci :L(V,JR)---+ L(V,JR2), i =1,2, V A E'11,T EL('1I, JR).
C1(T)(A)=(T(A), 0), C2(T)(A)=(0, T(Aằ,
(1) Show thatCi is a linear transformation fori=1,2.
(2) Show thatker(Ci )= {O},i= 1,2.
(3) Show that Im(C1)n Im(C2 ) = {O}.
(4) Let.4. = Im(Ci ),i =1,2and show that
17. Let '11be a vector space overJR. Acomplex structureon'11is a linear transformation
J :'11---+ V such that for all A in'11,
(1) Show that
J(x, y)=(-y,x) is a complex structure onJR2.
(2) Show that
J(x, y)=(y, -x) is a different complex structure onJR2.
18. For a real vector space0/with complex structureJ define a multi- plication of vectors in 0/by complex scalars by the fonnula
(a+bi)ãA= aA+bJ(A).
(1) Show that this definition of complex scalar multiplication makes 0/into a complex vector space.
(2) Show that onm3there is no complex structure.
19. A linear transfonnation T :0/~ 0/is called invertible if it is an isomorphism.
(1) If 0/is finite-dimensional, show that T is invertible if and only ifkerT= {O}.
(2) If 5, T are invertible, show that 5 . T is also. What is (5 . Ttl?
20. A sequence of real numbersao, al, ... , an," . is said to be eventu- ally zero if there is an integerk such thatai =0 for alli > k. Define
mooto be the set of all sequences(ao, at, ... , an, ... )that are eventually zero. Show that the usual addition of sequences and multiplication of sequences by a real number makesmoointo a vector space.
(1) Prove that p(m)andmooare isomorphic.
(2) Show thatp(m)is not isomorphic to .eo.
(3) Conclude that Theorem 8.6.3 need not be true in the infinite- dimensional case.
Chapter 10
Linear Transformations and Matrices
In this chapter we will begin the study of matrices, which are the analogue for linear transformations of coordinates for vectors. Our approach will be to consider first the case of a linear transformation
in some detail and then abstract the salient features to the general case.
This leads to an agebra for matrices, which we then pursue further.