7. The Elements of Vector Spaces
10.5 Exercises 151 11. Representing Linear Transformations
Did somebody just make a lucky guess? Perhaps, but a more logical development may be found in Chapter 13.
10.5 Exercises
1. For each of the following linear transformations of IR3to IR3 calcu- late the matrix relative to the standard basis:
(1) T : IR3---+ IR3byT(x, y,z)=(x+y+z,0, 0).
(2) Q :IR3---+ IR3 byQ(x, y, z) = (x, x+y, x+y +z).
(3) F:IR3---+ IR3byF(x, y, z)=(x+2y+3z, 2x+y, z - x).
(4) G :IR3---+ IR3byG(x, y, z)=(y - z, x +y,Z- 2x).
2. Calculate the matrices of the linear transformations T .Q :IR3---+ IR3
F . G :IR3---+IR3 Q .F . G :IR3---+ IR3 whereT,Q,F,and G are as in Exercise 1.
3. Suppose that the matrix of the linear transformation8 :IR3---+ IR3 is
[1 23] o1 -1 20 1
relativetothe standard basis of IR3 . Calculate 8(1, 2, 3).
4. Let P : IR3---+ IR3be the linear transformation given by P(x,y, z) = (x, y,O).
Calculate the matrix for P and for p2 relative to the standard basis of IR3.
5. Let 8, T : IR3---+ IR3be the linear transformations with matrices A and B relative to the standard basis of IR3•What is the matrix of 8 +T relative to this basis?
6. Let T : IR3---+ IR3be the linear extension of T(Et ) =(-1, 0, 2), T(E2 ) =(1, 1, -1), T(E3 )=(1, -3, 4).
Calculate the matrix of T relative to the standard basis.
7. Let T : IR2- > IR2 be the linear extension of T(1,1)=(1,-1),
T(1, -2)=(2, 2).
Calculate the matrix of the linear transformation T relative to the standard basis.
8. Let T : IR3- >IR3 be the linear transformation defined by T(x, y,z)=(0,y,z).
Calculate the matrix ofT relative to the standard basis ofIR3.
9. Let T : IR4- >IR4 be the linear transformation defined by T(x, y,Z,w)=(0,x,y,z).
Calculate the matrix ofT relative to the standard basis. Calculate the matrices ofT2, T3, T4 relativetothe standard basis.
10. Let T : IR3- >IR3 be the linear transformation whose matrix rela- tive to the standard basis is
[ ~2 -1 0i i].
(1) Calculate T(Et ), T(E2), T(E3).
(2) Letr be a real number. Calculate the matrix of rT relative to the standard basis.
11. Let T : IR2- >IR2 be the linear transformation whose matrix rela- tive to the standard basis is
(1) Calculate T(Et) and T(E2).
(2) Find At, A2 satisfying T(Ai )=Ei , i =1,2.
12. Let T : IR3- >IR3be the linear transformation whose matrix rela- tive to the standard basis is
[20-1]-1 11 1 0 .1
(1) Calculate T(Et), T(E2), T(E3), and T(1, 2,3).
(2) Is T swjective? (Check whether (1, 0, 0) belongs to Im(T).)
10.5 Exercises 153
j =1,2,3.
Let T : IR3~ IR3be the linear extension of the functionT(Ej)=Aj ,j = 1, 2, 3. Show that the matrix of T relative to the standard basis of IR3 is(ai,).
14. Let T :IR3~IR3be a linear transformation and A=(ai,j)its matrix with respect to the standard basis of IR3 . Regard the columns of A as vectors in IR3• Call these three vectorsA l,A2,~' Show that Im(T) = L({Al,A2,~}).
15. 1fT:IR3~ IR3is the linear transformation with matrix
[011]1101 0 1
with respect to the standard basis ofIR3,show that T is an isomorphism.
16. Perform the following matrix computations:
(a) 2 [3 1 4 -1] _ 3 [1 0 -1 7]
2 0 -1 2 -1 -2 0 -4 .
(b) 2 m-4m+8 [~]
17. Perform the following matrix multiplications:
(a) [1 2 3)ã m
[~ 1 ~] [n]ã
(b) 1
0
[1 0]
[~ 0 1 0 0 1 2 2
(c) 1 ° 1] ~ ~ . [2 2]ã
[~ 2 3] [0 ° 0]
(d) 5 6 . 1 0 0 .
8 9 0 0 0
[~ 0 0] [1 2 3]
(e) 0 o . 4 5 6 .
0 o 789
18. Which of the following matrices are nonsingular, involutory, idem- potent, nilpotent, symmetric, or skew-symmetric?
A = [1 -1] o 0' F = [-~ ~],
B = [~ :], G =[~ ~],
C =[ -1 l' 1-1] H =[-~ ~],
D = D n, J =[~ ~],
E = [-~ n, K =[~ ~].
Find the inverse for those that are invertible.
19. Show that a diagonal matrix
[ a1,1
A = 0 o
is nonsingular if and only if au, a22, a33~O. That is au ~0,a2,2~ 0,a3,3 ~O. If A has an inverse, what is it?
20. Show that a diagonal matrix
A= o
is nonsingular if and only ifallits diagonal entries are nonzero. Find its inverse.
21. If A=(ai,) is a matrix, we define the transpose of A to be the matrixAtr =(bi,), wherebi,}=a},i. Find the transpose of each of the following matrices:
[~ ~ ~], [1 2 3),
[~ ~ ~], [i:].
Show for any matrixA that(Atrtr =A.
10.5 Exercises 155 22. Let A be a square matrix, show that A is symmetric if and only if A=Atr,andAis skew-symmetric if and only ifA=-Atr.
23. For any square matrixAshow thatA+AIris symmetric andA - AIr is skew-symmetric.
24. Show every square matrix is the sum of a symmetric and a skew- symmetric matrix.
25. LetAandBbe 3 x 3 matrices. Show thatAIr Blr=(BA)'r. (Note that the order has been reversed.)
26. Show that the product of two lower triangular matrices is again lower triangular. If you cannot work the general case, do the 2 x 2 and 3x3cases.
27. Let A be an idempotent matrix. Show that1-A is also idempotent.
28. Show that a matrixAis involutory if and only if(I - A) .(I +A)=O.
29. Let A be a 3 x 3 matrix. Compute E(l,2) .A, A.E(l, 2), E(2, 1) . A, A.E(2, 1). What conclusion can you obtain in general for E(r,s)ãA and A . E(r,s)?
30. LetA=(ai,j) be a 3 x 3 matrix. ComputeA .E(r, s) and E(r,s)ãA.
31. A square matrix A is said to commute with a matrix B ifAB=BA.
When does a 3 x 3 matrix A commute with the matrix E(r,s)?
32. Show that if a 3 x 3 matrix A commutes with every 3 x 3 matrix B, then A is a scalar matrix. (HINT: If A commutes with every matrixB, it commutes with the 9 matricesEr,swherer,s=1,2,3. Use the result of the previous exercise.)
33. Find all 2 x 2 matrices that commute with
34. Construct a 3 x 3 matrixAsuch thatA3=I,butA2~I. (Tryto think of a simple linear transformationT :m3 --..m3withT3 =1and use its matrix relative to the standard basis.)
35. Let A be a 3 x 3 matrix and D the diagonal matrix
[
d1 0 0]
D= 0 d2 0 .
o 0 d3
(1) Compute Dã A.
(2) Compute A . D.
36. If A is an idempotent square matrix, show thatI - 2A is invertible.
(HINT: Idempotents correspond to projections. Interpret I - 2A as a reflection. Trythe 2 x 2 case first; then try to generalize.)
37. LetA, Band A+B be invertiblenxnmatrices. Show that A-I+B-1 is invertible. HINT: If you cannot think of anything else, try to show that
38. Find all the 2 x 2 matrices A satisfying the equation A2+I=O.
39. Find the inverse ofthe matrix
[ 0, 11]1 0 1 .
110 40. Show that
is invertible. Find its inverse. Solve the equation
forx andy.
41. Let A Emn•Write A for A regarded as annx 1 matrix. Show that A =0 if and only if the matrix product A . Air is the zero matrix.
42. A square matrix P is called a permutation matrix if in every column and every row there is exactly one nonzero entry, which is a 1.
Show:
(1) Iis a pennutation matrix.
(2) A skew-symmetric matrix is never a pennutation matrix.
(3) If A and Bare pennutation matrices, then so areAB and BA.
(4) Pennutation matrices are invertible, and the inverse is again a pennutation matrix.
(Ifyou have trouble with the general case, try the case of3 x 3 matrices.) 43. A square matrix P is called a generalized permutation matrix if in every column and every row there is exactly one nonzero entry.
Show:
(1) A diagonal matrix is a generalized pennutation matrix if and only if it is nonsingular.
(2) Give an example of a skew-symmetric generalized permuta- tion matrix.
10.5 Exercises 157 (3) IfA and B are generalized permutation matrices, then so are
ABand BA.
(4) Generalized permutation matrices are invertible, and the in- verse is again a generalized permutation matrix.
(Ifyou have trouble with the general case, try the case of3 x 3 matrices.)
Representing Linear
Transformations by Matrices
In this chapter we return to the ideas we introduced in Chapter 10 to represent a linear transformation T :rn3---+ rn3by a 3 x 3 matrix. There is, of course, nothing sacred aboutrn3 and its standard basis, and we will show that these ideas have far wider applications.