Linear Algebra

MATH 1071 Linear Algebra

Problems

Michaelmas 2012

We will (probably) not discuss all problems on the list while on the other hand during the term Iwill add a few more problems.

1 The vector space in Rn

1. Find the lengths of the two vectors and the angle between them in each of the following cases:

(i)(

12

),

(31

)in R2; (ii)

101

,

23−1

in R3;

(iii)

1111

,

−12−12

in R4; (iv)

120−2

,

2−120

in R4.

2. Find all the unit vectors in R4 which are orthogonal to1111

and

1−11−1

and make an angle of π/4 with

1000

.

3. Find the cross products of the following pairs of vectors:

(i)

−320

×1

70

; (ii)

125

× 3−17

; (iii)

327

×1

11

; (iv)

881

×5

52

.

1

1 THE VECTOR SPACE IN RN 2

4. Find the most general form for the vector u satisfying the equation

u×

211

=

100

×2

11

5. If a,b ∈ R3 with a 6= 0 show that the equation a× u = b has a solution if and only if a · b = 0

and find all the solutions in this case. [Hint: Before you start, ask yourself what sort of answeryou expect to get and what the set of solutions represents geometrically.]

6. Show that there are exactly two unit vectors in R3 which make an angle of π3 with both of the

vectors 122

,

21−2

.

Find the angle these two unit vectors make with each other.

7. Determine the orthogonal projection of each of the standard basis vectors of Rn onto the vectorsubspace defined by x1 + · · ·+ xn = 0.

8. Find the equation (in the form ax + by + cz = d) of the planes in R3 which contain the triplesof points:

(i)

101

,

111

,

22−1

; (ii)

−231

,

223

,

−4−11

.

9. Given a line ` in R3 and a point a not on ` show that there is a unique plane Π containing aand `. Find the equation of Π in the form ax+ by + cz = d when

a =

12−1

and ` is the line

x− 2 =y − 1

3=z − 2−2

.

10. If ` is a line in R3 and a is a point not on ` show that there is a unique plane Π through a whichmeets ` orthogonally. Find the equation of Π in the form ax+ by + cz = d when

a =

12−1

and ` is the line

x− 12

=y + 1

2= z.

1 THE VECTOR SPACE IN RN 3

11. Write down the equations for the line in R3 through the point

a =

124

parallel to the line

x− 1 =y + 5

2=z

2.

Find the distance between these lines.

12. Find the distance between the lines `1 and `2 in R3 when:

(i) `1 :x− 2

3=y − 5

2=z − 1−1

, `2 :x− 4−4

=y − 5

4= z + 2;

(ii) `1 : x = y = z, `2 : 2x− 1 = y + 1 = 2z + 1.

13. If `1, `2 are non-parallel lines in R3 show there is a unique line `3 which meets `1 and `2 orthog-onally. Find `3 when `1, `2 are given by

`1 : x− 2 =y − 1

3=z − 3−2

, `2 : x+ 1 = y + 2 = z + 3.

14. If ` is a line in R3 and a is a point not on ` show that there is a unique line `′ through a whichmeets ` orthogonally. Find `′ when

a =

12−1

and ` is the line

x+ 2 =y − 1

2=z + 1

2.

15. Determine all lines which meet the lines

`1 :x− 1

2=y

2= z − 1, `2 : x− 1 =

y + 1−2

=z

2

at an angle of π3 . What do you notice about the configuration formed by these lines?

16. Let ` be a line in R3 not passing through the origin. Show that there is a unique line throughthe origin that intersects ` orthogonally. What goes wrong if ` passes through the origin?

17. (i) Show that every pair of opposite edges of a regular tetrahedron is orthogonal.

(ii) Show that if two pairs of opposite edges of a tetrahedron are orthogonal then the third pairof opposite edges are also orthogonal.

(iii) Is it true that if every pair of opposite edges of a tetrahedron is orthogonal then thetetrahedron is regular? Give either a proof or a counterexample.

2 SYSTEMS OF LINEAR EQUATIONS 4

2 Systems of linear equations

18. Find the solution sets to the following (nonlinear!) equations or systems of equations (with xand y real):

(i){y = x2, (x− 1)2 + y2 = 1.

}. ;

(ii) {2x+ 3y = 2, 4x+ 6y = 8} . ;(iii) 1− 2x =

√x2 + 3x+ 7;

(iv) sinx = 1/√

2;

(v) y = x2 + 1.

19. Solve the following systems of linear equations:

(i)2x1 +x2 +3x3 = 0,3x1 −2x2 +x3 = 0,x1 −3x2 −2x3 = 1.

(ii)x1 +x2 −x3 = 7,

4x1 −x2 +5x3 = 4,2x1 +2x2 −3x3 = 0.

(iii)x1 +x2 −x3 = 7,

4x1 −x2 +5x3 = 4,6x1 +x2 +3x3 = 18.

(iv)2x1 −x2 −x3 = 0,x1 +x2 +2x3 = 0,

7x1 +x2 −3x3 = 0,2x2 −x3 = 0.

(v)x1 + x2 − x3 − x4 = −1,

3x1 +4x2 − x3 −2x4 = 3,x1 +2x2 + x3 = 5.

(vi)x1 −2x2 + x3 + x4 = 2,

3x1 +2x3 −2x4 = −8,4x2 − x3 − x4 = 1,

5x1 +3x3 − x4 = 0.

2 SYSTEMS OF LINEAR EQUATIONS 5

(vii)x2 +x3 + · · · +xn−1 +xn = 1,

x1 +x3 + · · · +xn−1 +xn = 2,...

......

...x1 + · · · +xr−1 +xr+1 + · · · +xn = r,

......

......

x1 +x2 +x3 + · · · +xn−1 = n.

20. Find the conditions on a, b, c such that the system of linear equations

2x1 +3x2 − x3 = a,x1 −x2 +3x3 = b,

3x1 +7x2 −5x3 = c.

is consistent and find the set of solutions in this case.

21. Prove that the system of linear equations

x1 +2x2 +3x3 −3x4 = k1,2x1 −5x2 −3x3 +12x4 = k2,7x1 +x2 +8x3 + 5x4 = k3.

has a solution if and only if 37k1+13k2−9k3 = 0. Find the solutions when k1 = 1, k2 = 2, k3 = 7.

22. For which values of t does the system of linear equations

tx1 + x2 + x3 = 1,x1 +tx2 + x3 = 1,x1 + x2 +tx3 = 1.

have (a) a unique solution; (b) infinitely many solutions; (c) no solution? Find the solutions incases (a) and (b).

23. Use Maple to solve the equations

x− 2y + 3z = v1,

3x− 2y + z = v2,

−x+ 2y + 4z = v3

when

(i) v =

v1v2v3

=

121

, (ii) v =

v1v2v3

=

12−1

.

Use Maple to produce a plot of the planes given by the equations in (i) and (ii) where the pointof intersection can be clearly seen.

3 MATRICES 6

3 Matrices

24. Compute each of the following matrix products:

(i)(−3 1 0

0 2 5

) 1 1 00 −2 1−4 1 −3

; (ii)

1 1 10 1 −11 0 2

−1 1 −12 −6 −1−7 0 1

;

(iii)

1−12

(3 0 −1)

; (iv)

2 01 −30 4

( 2 1 0−3 0 5

);

(v)(

2 1 0−3 0 5

) 2 01 −30 4

.

25. Find a, b, c, d such that (2 14 2

)(a bc d

)=(

1 32 6

).

Do there exist p, q, r, s such that (p qr s

)(2 14 2

)=(

1 32 6

)?

26. Compute 1 a c0 1 b0 0 1

1 x z0 1 y0 0 1

.

Hence show by induction that1 1 00 1 10 0 1

n

=

1 n n(n− 1)/20 1 n0 0 1

.

27. Let A,B,C be matrices of sizes m × p, p × q, q × n respectively. Find expressions (in termsof m,n, p, q) for the number of times matrix entries are multiplied together in the computa-tions of (AB)C and A(BC). If m > p > q > n which of (AB)C and A(BC) requires fewermultiplications?

28. Show that an n× n matrix A commutes with every n× n matrix if and only if it A is a scalarmatrix, that is to say A = kI for some scalar k.

Hint: Consider AE and EA, where E is an elementary matrix.

29. Determine which of the following matrices are invertible and find the inverse when it exists.

(i)(

1 23 4

), (ii)

(1 22 4

), (iii)

(1 n0 1

), (iv)

(1 1−1 1

),

(v)(

0 −11 0

), (vi)

(cos θ − sin θsin θ cos θ

), (vii)

(cos θ sin θsin θ − cos θ

).

3 MATRICES 7

30. Determine which of the following matrices are invertible and find the inverse when it exists.

(i)

2 1 03 2 14 3 2

, (ii)

2 3 41 2 30 1 2

, (iii)

2 1 01 2 10 1 2

, (iv)

1 a c0 1 b0 0 1

,

(v)

2 −1 21 −2 −22 2 −1

; (vi)

2 −1 −22 −2 −21 2 −1

; (vii)

2 1 11 2 11 1 2

.

31. Find the inverses of the following matrices:

(i)

1 − 3 0 −23 −12 −2 −6−2 10 2 5−1 6 1 3

(ii)

2 1 0 01 2 1 00 1 2 10 0 1 2

(iii)

1 −1 1 −10 1 0 11 0 −1 00 1 0 −1

(iv)

0 1 · · · 1 11 0 · · · 1 1...

.... . .

......

1 1 · · · 0 11 1 · · · 1 0

(v)

1 2 · · · n− 1 n0 1 · · · n− 2 n− 1...

.... . .

......

0 0 · · · 1 20 0 · · · 0 1

32. If A is an n× n matrix such that A2 + 2A+ 3I = 0 show that A is invertible and express A−1

as a polynomial in A.

33. Let A be an n× n matrix such that Ar = 0 for some positive integer r (such a matrix is callednilpotent). Show that I + A is invertible and express the inverse as a polynomial in A. Deducethat, under the same assumptions on A, any polynomial in A with nonzero constant term isinvertible.

34. An n × n matrix A = (aij) is upper triangular if aij = 0 for i > j. Show that the inverse of aninvertible upper triangular matrix is upper triangular.

35. Show that the inverse of an invertible symmetric matrix is symmetric. Is the inverse of aninvertible skew-symmetric matrix skew-symmetric?

36. Show that every n × n matrix A can be written in the form A = B + C where B is symmetricand C is anti-symmetric. If A = B′ + C ′ is another such decomposition show that B = B′ andC = C ′.

37. Let n be an integer, not necessarily positive. Find all 2 × 2 matrices B (with real coefficients)such that Bn = A when :

(i) A =(

1 00 3

), (ii) A =

(1 10 1

), (iii) A =

(−1 −2

4 5

), (iv) A =

(2 1−1 0

).

3 MATRICES 8

Hint: For (i) and (ii) use the fact that, for any such B, BA = Bn+1 = AB. For (iii) and (iv)consider PAP−1 where

P =(

2 11 1

)and then use (i) and (ii).

38. For each n ≥ 4 Give an example of an n × n matrix A such that An = 0 but An−1 6= 0. (Trythe cases n = 2 and n = 3 first.)

39. If A = (aij) is an n× n matrix the trace of A is defined by

tr(A) =n∑i=1

aii.

Show that if A and B are n×n matrices then tr(AB) = tr(BA) and deduce that if B is invertiblethen tr(BAB−1) = tr(A).

40. An n × n matrix A is strictly upper triangular if, for all i ≥ j, the (i, j)-entry aij = 0. Provethat:

(a) the product of any two strictly upper triangular matrices is strictly upper triangular;

(b) the product of any three strictly upper triangular 3× 3 matrices is zero.

Formulate and prove the result corresponding to (b) for strictly upper triangular n×n matrices.

41. (i) Suppose that A is an n×n matrix and (during, for example, an attempt to find the inverseof A) the matrix (A | In) has been transformed by EROs into the matrix (A′ | B). Usingthe fact that EROs may be performed by multiplying on the left by certain matrices, orotherwise, show that A′ = BA.(This provides an alternative proof that the inversion algorithm works and also proves amethod of checking, in the middle of such a calculation, that no error has crept in.)

(ii) Show that if AB is a product of matrices then the result of performing a certain ERO onAB is the same as that of performing this ERO on A and then multiplying the result (onthe right) by B.

4 DETERMINANTS 9

4 Determinants

42. Compute the following determinants:

(i) det

1 3 42 0 11 3 2

, (ii) det

5 3 12 0 01 2 0

, (iii) det

x y zy z xz x y

,

(iv) det

x 1 11 x 11 1 x

, (v) det

3 −1 2 41 1 0 3−2 4 1 5

6 −4 1 2

, (vi) det

x y z ty z t xz t x yt x y z

,

(vii) det

x 1 0 01 x 1 00 1 x 10 0 1 x

.

In the case of (iii) and (vi) express the determinant as a product of linear factors.

Hint: To do this you will need to use complex 3rd and 4th roots of unity respectively. Forexample, the answer to (iii) is −(x+ y + z)(x+ ωy + ω2z)(x+ ω2y + ωz), where ω = e2πi/3.

43. Show that the determinant of the n× n matrixx a · · · a aa x · · · a a...

.... . .

......

a a · · · x aa a · · · a x

is (x+ (n− 1)a)(x− a)n−1.

44. Express the determinant of the n× n matrix

x 0 0 · · · 0 0 −an1 x 0 · · · 0 0 −an−1

0 1 x · · · 0 0 −an−2...

......

. . ....

......

0 0 0 · · · x 0 −a3

0 0 0 · · · 1 x −a2

0 0 0 · · · 0 1 x− a1

as a polynomial in x.

45. For which values of λ is the matrix λI −A singular when A is

(i)(

1 21 −1

), (ii)

0 1 11 0 11 1 0

, (iii)

0 1 21 0 12 1 0

?

4 DETERMINANTS 10

46. Comparative efficiency of methods for calculating determinants:

(a) Find the number of arithmetic operations (counting each multiplication and addition asa separate operation) needed to evaluate a 3 × 3 determinant by, on the one hand, usingrow operations to reduce to echelon form and, on the other hand, expanding by a row orcolumn.

(b) Find a corresponding estimate for the number of arithmetic operations needed for eachmethod in the n× n case.

47. Show that the quadrilaterals with the following vertices are parallelograms and find their areas:

(i)(−1−2

),

(0−1

),

(22

),

(11

); (ii)

(−2−1

),

(15

),

(39

),

(03

).

48. Find the area of the triangles with vertices:

(i)(

13

),

(24

),

(−12

); (ii)

(4−2

),

(−24

),

(−11

).

49. Find the scalar triple products of the following sets of vectors:

(i)

1

11

,

11−1

,

−111

; (ii)

3

21

,

735

,

24−1

; (iii)

−2

10

,

234

,

044

.

50. Find the volume of the parallelepiped with vertices−1−1−1

,

0−11

,

10−2

,

−120

,

200

,

022

,

13−1

,

231

.

51. Find the volume of the tetrahedron with vertices

a =

123

, b =

420

, c =

252

, d =

13−1

.

5 SPANNING SETS AND LINEAR INDEPENDENCE 11

5 Spanning sets and linear independence

52. Determine whether or not each of the given sets of vectors is linearly independent :

(i)

−3

42

,

7−1

3

,

128

, (ii)

−3

42

,

7−1

3

,

118

,

(iii)

1021

,

−1−2

4−3

,

2113

, (iv)

1031

,

2103

,

1−1

70

,

31−5

2

,

47−1

3

,

(v)

3142

,

−1−2

33

,

1012

,

11−4

1

, (vi)

2112

,

1221

,

−1−1

11

,

031−2

.

53. For what values of t is each of the following subsets of R3 linearly independent?

(i )

1

23

,

2−1

4

,

3t4

(ii )

2−3

1

,

−46−2

,

t12

.

54. Identify which of the following are bases of R3:

(i)

0

12

,

113

,

−101

; (ii)

0

12

,

113

,

101

;

(iii)

2

10

,

121

,

012

; (iv)

1

00

,

110

,

111

.

55. Consider the vectors:

a =

111

, b =

1−12

, c =

241

, d =

15−1

, e =

123

, f =

000

Which of the following subsets of R3 is (a) linearly independent, (b) a spanning set, (c) a basis?

(i) {a,b}, (ii) {a, c}, (iii) {a,b, c}, (iv) {a,b, c,d},(v) {a,b, e}, (vi) {a,b,d, e}, (vii) {a,b, e, f}.

56. Which of the following subsets of R4 are (a) linearly independent, (b) a spanning set, (c) a basis?

(i)

1023

,

−1310

,

012−1

,

1−121

, (ii)

1023

,

−1310

,

012−1

,

0212

,

3445

.


57. Show that

11011

,

01101

,

10101

is a linearly independent subset of R5 and find a basis of R5 which contains it.

58. Show that if a, b, c are vectors in R3 then{b + c, c + a, a + b

}is a basis of R3 if and only if{

a, b, c}

is.

59. If a, b, c are vectors in R3 the show that{b × c, c × a, a × b

}is a basis of R3 if and only if{

a, b, c}

is.

60. If {(ab

),

(cd

)}is a basis for R2 express (

10

)and

(01

)as linear combinations of (

ab

)and

(cd

).

61. Show that 0

21

,

750

,

31−1

is a basis for R3 and find the coordinates of the vectors 7

130

and

132

with respect to this basis.

62. Show that

0111

,

1011

,

1101

,

1110

is a basis for R4 and find the coordinates of the vectors1234

and

−12−12



63. Show that 2i

10

,

2−11

,

01 + i1− i

is a basis for C3 and find the coordinates of the vectors1

00

,

010

and

001


64. Write down the dimension of the subspace of R5 of solutions each of the following systems oflinear equations and check your answer by finding a basis in each case:

(i)4x1 +x3 −x5 = 0,

x2 3x4 = 0,x4 +5x5 = 0.

(ii)x1 −x2 +x3 −x4 +x5 = 0,

x2 +x3 +x4 +x5 = 0,x3 +x4 −x5 = 0,

x4 +4x5 = 0.

(iii)3x1 +2x2 −x3 +2x4 −x5 = 0,5x1 −x2 +3x3 −x4 +x5 = 0.

(iv)x1 + x2 + x3 + x4 + x5 = 0.

65. If a1,a2 ∈ Rn show that:

(i) any subset of Span{a1,a2} containing three elements is linearly dependent;

(ii) if b1,b2 are linearly independent vectors in Span{a1,a2} then Span{a1,a2} = Span{b1,b2}.

66. Let a1,a2,a3 ∈ R2 be three noncollinear points. Show that every v ∈ R2 may be uniquelyexpressed in the form

v = λ1a1 + λ2a2 + λ3a3, λ1 + λ2 + λ3 = 1, λ1, λ2, λ3 ∈ R.

Draw the set of points v ∈ R2 for which λ1, λ2, λ3 ≥ 0 when

a1 =(

10

), a2 =

(−12

), a3 =

(−1−2

),

6 OTHER REAL VECTOR SPACES 14

and find λ1, λ2, λ3 when

v =(−1

30

).

6 Other real vector spaces

67. Show that the following pairs of functions are linearly independent in the vector space of allfunctions from R to R:

(a) 1, t; (b) t, t2; (c) et, t; (d) tet, e2t; (e) sin(t), cos(t); (f) sin(t), sin(2t).

68. Show that for any integer n ≥ 1 the functions

1, cos t, sin t, . . . , cosnt, sinnt

are linearly independent in the vector space of question 67.

69. Let r be a fixed real number. Show that {1, x + r, (x + r)2} is a basis for R[x]2 and find thecoordinates of a0 + a1x+ a2x

2 with respect to this basis.

70. The k-th Legendre polynomial Pk(x) is defined by

Pk(x) =1

2kk!dk

dxk(x2 − 1)k.

Show that, for each non-negative integer n, the set {P0(x), . . . , Pn(x)} is a basis for R[x]n.

71. Show that the subset of R[x]n, (n ≥ 4), of those polynomials whose graphs touch the x-axistangentially at x = 0 and x = 1 is a vector subspace and find its dimension. Also find a basisfor this subspace. (If you cannot do the case of general n, try n = 4, 5 first.)

72. Show that the subset{f(x) ∈ R[x]n|f(0) = f(1) = 0, f ′(0) = 1, f ′(−1) = −1

}is an affine subspace but not a vector subspace of R[x]n.

(An affine subspace of a vector space V is a subset of the form

a + U = {a + u |u ∈ U},

where U is a vector subspace of V .)

73. (Magic squares) Let M(d), (d ∈ R) be the subset of M3(R) consisting of those 3 × 3 matri-ces such that the sum of the elements in each row, each column, on the diagonal and on thecodiagonal is d. (An element of M(d) is called a magic square with magic number d.)

(i) Show that M(0) is a vector subspace of M3(R) and find a basis for M(0).

6 OTHER REAL VECTOR SPACES 15

(ii) Show that

M(d) =d

3

1 1 11 1 11 1 1

+M(0).

(iii) Find all magic squares whose entries are the integers 1 to 9.

74. Let W1 and W2 be the subsets of the vector space M2(R) of 2 × 2 matrices with real coeffi-cients consisting of symmetric and skew-symmetric matrices respectively. Show that W1,W2 aresubspaces of M2(R) and find bases for them. Show also that M2(R) = W1

⊕W2.

75. Show that the vector space of all symmetric n× n-matrices has dimension n(n+ 1)/2. What isthe dimension of the vector space of all skew-symmetric n× n-matrices?

76. Let E be the 2× 2 elementary matrix (1 10 1

)and let W1, W2, W3 be the vector subspaces of M2(R) defined by

W1 = {A ∈M2(R)|AE = EA},W2 = {A ∈M2(R)|AtE = EAt},W3 = {A ∈M2(R)|A = At}.

Determine the dimensions of W1 +W2, W1 ∩W2, (W1 +W2) ∩W3 and W1 ∩W2 ∩W3 and finda basis in each case.

77. Decide which of the following subsets of R[x]3 are linearly independent, which are spanning setsand which are bases:

(a) {1, 1 + x, 1 + x+ x2};(b) {1, 1 + x, 1 + x+ x2, 1 + x+ x2 + x3};(c) {1 + x, x+ x2, x2 + x3, 1 + x3, 1 + x+ x2};(d) {1 + x− x3, −1 + x+ x2, −x+ x2 + x3, 1− x+ x3}.

78. Decide which of the following subsets of M2(R) are linearly independent, which are spanningsets and which are bases:

(a) {(1 00 0

),

(1 10 0

),

(1 11 0

)};

(b) {(1 00 0

),

(1 10 0

),

(1 11 0

),

(1 11 1

)};

7 LINEAR MAPS 16

(c) {(1 00 1

),

(0 11 0

),

(1 10 0

),

(0 01 1

)};

(d) {(1 10 1

),

(1 01 1

),

(1 00 −1

),

(0 11 0

)}.

79. If C∞(R) is the vector space of infinitely differentiable functions f : R→ R show that the subset{ex, e2x, e3x} of C∞(R) is linearly independent.

(Hint: Given a linear relation λ1ex + λ2e

2x + λ3e3x = 0 differentiate this twice with respect

to x to get two other linear relations. What can you say, for each given value of x, about thedeterminant of this system of 3 linear equations in 3 unknowns?)

80. If a1, . . . , an are distinct real numbers show that the functions ea1x, . . . , eanx are linearly inde-pendent in the vector space C∞(R).

7 Linear Maps

81. Give a sketch showing the image of the unit square in R2 with vertices at the points(00

),

(10

),

(01

),

(11

)under each of the following linear transformations T : R2 → R2 given by T (x) = Ax where:

(a) A =(

2 00 3

), (b) A =

(1 20 1

), (c) A =

(1 22 4

),

(d) A =(

cos θ − sin θsin θ cos θ

), (e) A =

(cos θ sin θsin θ − cos θ

), (f) A =

(2 11 2

),

(g) A =(

1 22 1

).

82. Find the vectors obtained from 211

and

11−1

by reflection in

(a) the plane x1 − 2x2 + 2x3 = 0;

(b) the plane x1 − 2x2 + 2x3 = 1.

[Hint: For (b) first apply a translation (to all the data) to get a plane through the origin, do thecalculation for that plane, and then translate back again.]

7 LINEAR MAPS 17

83. Find the matrix A of the linear map of R3 to itself given by reflection in the plane 2x−y+2z = 0.Verify that A2 = I.

84. Find the matrix A of the linear map of R3 to itself given by orthogonal projection onto the planex+ y + z = 0. Verify that A2 = A.

85. Determine the matrices A,B of the linear map of R3 to itself given by rotation through ±π3

about the line 2x = y = z. What do you notice about the way A is related to B? Check thatA,B are orthogonal matrices and compute A2 and B2.

86. If T : R3 −→ R3 is the linear transformation defined by

T

xyz

=

y + zz + xx+ y

,

find the image under T of each of the following:

(a) the line x = y = z;

(b) the plane x+ y + z = 0.

Use your answer to describe T geometrically and find T−1.

87. Determine the matrix A of the linear map of Rn to itself given by reflection in the planex1 + . . .+ xn = 0 and check that A is symmetric and orthogonal and that A2 = I.

88. If A is a 2× 2 matrix such that An = 0 for some positive integer n, prove that A2 = 0.

89. If A is a 2× 2 matrix such that A2 = 0 but A 6= 0, prove that there is an invertible 2× 2 matrixP such that

P−1AP =(

0 10 0

).

90. If A is a 2 × 2 matrix such that A2 = A but A 6= 0, I, prove that there is an invertible 2 × 2matrix P such that

P−1AP =(

1 00 0

).

91. Let T : R[x]3 → R be the function defined by

T (p(x)) =∫ 1

0p(x)dx, p(x) ∈ R[x]3.

Show that T is a linear map and find a basis for ker(T ).

92. Show that the following functions T : R[x]n → R[x]n are all linear transformations:

(i) T (p(x)) = p(x+ 1),

7 LINEAR MAPS 18

(ii) T (p(x)) = p′(x),

(iii) T (p(x)) = p′′(x) + 2p′(x) + p(x).

Find the image and kernel of T in each case and find a basis of ker(T ) in each case.

93. Let A be an m× n matrix, Q a non-singular m×m matrix and P a non-singular n× n matrix.Show that r(QAP ) = r(A).

94. Find the ranks of the following matrices:

(i)

2 −1 3 01 4 −3 93 1 2 5

, (ii)

−1 0 0 1 0

0 1 0 0 −11 0 −1 0 00 −1 0 1 00 0 1 0 1

, (iii)

α β γβ + γ γ + α α+ ββγ γα αβ

.

(consider all possible cases).

95. For each real number λ find the rank of

(i)

λ− 1 1 11 λ− 1 11 1 λ− 1

, (ii)

λ 0 1 00 λ 0 1−1 0 λ 0

0 −1 0 λ

.

96. Prove by induction on n that the rank of a skew-symmetric n× n matrix is even.

97. Let A be an m× n matrix, B an n× p matrix. Prove that r(AB) ≤min(r(A), r(B)).

98. Let T : R3 → R2 and S : R2 → R3 be linear transformations. Show that the compositeST : R3 → R3 is not invertible. Give an example of S, T for which TS : R2 → R2 is invertible.

99. If

A =(a bc d

)and A2 = A, show that r(A) = a+ d.

100. Find the rank and nullity of the linear transformation T : R3 → R3 when:

(i) T (x, y, z) = (x+ y + z, x+ y + z, x+ y + z);

(ii) T (x, y, z) = (y + z, z + x, x+ y);

(iii) T (x, y, z) = (y − z, z − x, x− y);

(iv) T (x, y, z) = (y + z − 2x, z + x− 2y, x+ y − 2z).

101. For each λ ∈ R find the rank and nullity of the linear transformation Tλ : R[x]n → R[x]n definedby

Tλf(x) = (1− x2)f ′′(x)− 2xf ′(x) + λf(x).

8 CHANGE OF BASIS 19

102. (a) Give an example of a linear transformation T : V → V with ker(T ) ∩ im(T ) 6= {0}.(b) Show that if T : V → V is a linear transformation satisfying T 2 = T then we have

ker(T ) ∩ im(T ) = {0}.

103. A linear transformation P : V → V of a vector space V is called a projection if P 2 = P . Showthat

(i) if P : V → V is a projection then I − P is a projection and V = im(P )⊕

ker(P );

(ii) if P : V → V is a projection, where V is a real vector space of finite dimension n, then thetrace of the matrix of P with respect to any basis of V is equal to the rank of P ;

(iii) if P1, P2 : V → V are projections then P1+P2 is a projection if and only if P1P2 = P2P1 = 0.

8 Change of basis

104. Let T : R3 → R2 be a linear transformation whose matrix with respect to bases {e1, e2, e3} ofR3 and {f1, f2} of R2 is

A =(

1 0 −32 1 −1

).

Find the matrix of T with respect to bases {e1, e2, e3} of R3 and {f1, f2} of R2 given bye1 = e1 + e2,e2 = e2 + e3,e3 = e1 + e3

,

{f1 = f1 − 2f2,f2 = f1 + f2

}.

105. Let T : R2 → R3 be given with respect to the standard bases of R2,R3 by the matrix

A =

1 22 01 4

.

Find bases of R2,R3 with respect to which T is given by a matrix of the form(Ir 00 0

)where r is the rank of T and 0 stands for a matrix of zeros of the appropriate size (possiblyzero).

106. Find non-singular real matrices P and Q such that QAP has the canonical form(Ir 00 0

)

8 CHANGE OF BASIS 20

when

(a) A =

1 1 2−1 −3 8

4 −3 −71 12 −3

, (b) A =

1 2 3 40 1 −1 53 4 11 2

.

107. Find bases of R3,R4 with respect to which the linear transformation T : R3 → R4 defined by

T (x, y, z) = (x− y + z, 2x+ y, 3y − 2z, 3x+ 3y − z)

has matrix of the form (Ir 00 0

).

108. Let T : R3 → R3 be the linear transformation defined by

T

xyz

=

3x+ z−2x+ y

−x+ 2y + 4z

.

Find the matrix which represents T with respect to:

(a) the standard basis in both copies of R3;

(b) the basis consisting of 101

,

−121

,

211

in both copies of R3.

Show that T is invertible, and give an explicit formula for T−1 in each case.

Linear Algebra

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