problems in advanced engineering mathematics

Problems in AEM Section 0 p. 1

Keith Anguige (with thanks to Peter Furlan)

Problems inAdvanced Engineering Mathematics

November 9, 2015

This is a collection of exercises for the course Advanced Engineering Mathematics.

Most of the homework problems will be taken from here. More generally, it will serve as a source of practice

problems for the examination.


Contents

1 Vectors 3

2 Matrices 4

3 Equation systems 7

4 Vector Spaces 8

5 Linear mappings 9

6 Scalar product 13

7 Eigenvalues and Jordan-decomposition 15

8 Quadratic forms 19

9 Numerics of equation systems 20

10 Interpolation 21

11 Best-fit problems 22

12 Normed spaces 23

13 Matrix decompositions 24

14 Fundamentals of differential equations 28

15 Linear differential equations with constant coefficients 29

16 Differential equation systems 30

17 Differential equation systems with constant coefficients 31

18 Derivatives 33

19 Taylor series 35

20 Extreme values 36

21 Laplace transform 37


1 Vectors

Problem 1.1

(i) Show that ~a1 = (1, 1, 1)T , ~a2 = (1,−1, 0)T , ~a3 = (1, 1,−2)T form a basis of R3. Write (1,−2, 5) inthe form u1~a1 + u2~a2 + u3~a3

(ii) Determine whether the following pairs of vectors are bases of R2:

a) (3, 1)T , (2, 4)T

b)(− 1√

2,√2)T

, (2,−4)T

Problem 1.2

Are these vectors linearly independent? If not, represent one of the vectors as a linear combination of theothers.

1001

,

2−6−40

,

10−210

,

3112

.

Problem 1.3

Which of these sets are subspaces?

(i) {(0, 0, 0)} ∈ R3,

(ii) {(1, 1, 1)} ∈ R3,

(iii) {(x1, x2) |x2 = x22} ∈ R2,

(iv) {(x1, x2) |x31 = x32} ∈ R2,

(v) {(x1, x2, x3) |x2 ≥ 0} ∈ R3.


2 Matrices

Problem 2.1

Let

A =

[1 1 −10 2 1

]and B =

2 −11 −30 2

.

Calculate (whenever well-defined) A · B, B · A, A2, B−1, (A · B)−1 and (B · A)−1, and determine the rankof all resulting matrices.

Problem 2.2

Let En be the n × n identity matrix, A an invertible m×m matrix, B an (m× n matrix and C an n×mmatrix.

(i) Show for n = 3, m = 2, A =

[1 22 3

], C =

1 04 11 −1

and B =

[2 0 00 1 −1

]:

If W = En + C(A−BC)−1B exists, then W = En − CA−1B is invertible, with W−1 = W .

Hint: Calculate W , W and WW.

(ii) Show that WW = En is always true.

Problem 2.3

(i) Find the inverse of A =

1 a 0 00 1 a 00 0 1 a0 0 0 1

.

(ii) Find the inverse of B =

[1 32 5

].

(iii) Use block matrices to find the inverse of C =

1 0 1 3 0 0 0 00 1 2 5 0 0 0 00 0 1 0 1 3 0 00 0 0 1 2 5 0 00 0 0 0 1 0 1 30 0 0 0 0 1 2 50 0 0 0 0 0 1 00 0 0 0 0 0 0 1

Problem 2.4

Let A =

1 32 51 1

.

Are there matrices B or C so that BA = E2 resp. AC = E3?

If yes, find all such matrices.

Hint: BA = E2 ⇔ A⊤B⊤= E2

Problem 2.5

Use the Gauß algorithm to reduce A =

−1 2 1−1 1 5−2 4 3

to the identity matrix.


Write down which matrix-multiplications of type C(i, j;α), D(k;α) or F (k, l) yield the transformation.

Problem 2.6

Find all orthogonal matrices of the form A =

1/2 ∗ ∗∗ ∗ 1∗ ∗ ∗

.

Problem 2.7

Let A =

1 2 3 40 1 2 30 0 1 20 0 0 1

. Find the inverse of A by substituting B = A−1 =

a b c d0 e f g0 0 h i0 0 0 j

into AB = E4.

Problem 2.8

Let A, B be symmetric, U , V orthogonal, r ∈ R.

Prove or give a counter-example to

(i) AB symmetric

(ii) A2 symmetric

(iii) A−1 symmetric

(iv) A+B symmetric

(v) rA symmetric

(vi) UV orthogonal

(vii) U2 orthogonal

(viii) U−1 orthogonal

(ix) U + V orthogonal

(x) rU orthogonal

Problem 2.9

Let A be an invertible n× n-matrix, ~v1, . . . , ~vk ∈ Rn.

Prove ~v1, . . . , ~vk linearly independent ⇔ A~v1, . . . , A~vk linearly independent.

Conclude that the rank of BA is the rank of B for any matrix B of appropriate size.

Problem 2.10

(i) Prove that

a) the inverse of an upper-triangular matrix is again upper triangular.

b) the product of two upper-triangular matrices is upper triangular.

(ii) What can one say about the powers of upper-triangular matrices with all diagonal elements equal to

zero? Use A =

0 1 1 10 0 2 20 0 0 30 0 0 0

as an example to find a general rule.

Problem 2.11

(Block matrices)


(i) Let A11, A21 and A22 be n× n-matrices, A11 and A22 invertible. 0 denotes the n× n zero matrix.

Find the inverse of A =

[A11 0A21 A22

]with help of the approach A−1 =

[B11 B12

B21 B22

]and AA−1 = E2n.

(ii) Let A =

1 0 0 0 0 0 0 00 1 0 0 0 0 0 01 −2 1 0 0 0 0 02 −3 0 1 0 0 0 00 0 1 −2 1 0 0 00 0 2 −3 0 1 0 00 0 0 0 1 −2 1 00 0 0 0 2 −3 0 1

. Find the inverse of A. First divide A into 16 blocks of size

2× 2 and compute the inverse of a block matrix of this form.

Problem 2.12

(i) Show that there is no 2× 2-matrix B so that AB = A⊤holds for every 2× 2-matrix A.

(ii) Let A =

0 1 01 0 00 0 1

. Find all matrices B with AB = BA


3 Equation systems

Problem 3.1

Find the solution ofx1 − x2 + 4x3 + 6x4 = 0x1 + 2x2 − 2x3 − 3x4 = 6x1 + x2 = 4

Problem 3.2

Let α ∈ R.x1 + 2x2 − 4x3 = αx1 − 3x2 + x3 = 12x1 + 5x2 − 9x3 = 2

(i) Find all solutions of the homogeneous system.

(ii) For which α ∈ R is the inhomogeneous system solvable? What are the solutions?


4 Vector Spaces

Problem 4.1

Let U = {~x ∈ R3 | x1 + 2x2 − x3 = 0} and V = {~x ∈ R

3 | x1 − 2x2 + 2x3 = 0}.Find a basis of

(i) U

(ii) V

(iii) U ∩ V

(iv) span U ∪ V


5 Linear mappings

Problem 5.1

The quadrangle V ⊂ R2 has the corners (1,0), (1,1), (0,1) and (-1,0).

Sketch the image of V under the linear maps given by the matrices

A1 =

[2 00 1

], A2 =

[−1 00 −1

], A3 =

[3 00 −1

]

A4 =

[2 10 2

], A5 =

[2 −11 2

], A6 =

[0 −1

−1 0

].

Problem 5.2

Let ~x1 =

101

, ~x2 =

220

and ~x3 =

111

.

(i) Show that {~x1, ~x2, ~x3} is a basis of R3.

(ii) Find the linear map with matrix A such that A~x1 =

314

, A~x2 =

442

and A~x3 =

434

.

(iii) Calculate A~x4 and A~x5, where ~x4 =

010

and ~x5 =

432

(iv) Confirm A = B · S−1. Here S is the matrix with columns ~x1, ~x2, ~x3 and B the matrix with columnsA~x1, A~x2, A~x3.

Problem 5.3

(i) Find matrix representations of these linear maps L : R2 → R2:

a) L is a rotation about the origin by the angle α.

b) L is inversion through the origin

c) L is a reflection about the x-axis

d) L is a reflection about the straight line ~x · ~n = 0.

(ii) Let U ∈ R3 be the span of the two basis vectors ~u1 =

1−22

and ~u2 =

221

. What is the matrix of

the orthogonal projection P : R3 → U?

Problem 5.4

Let

~u1 =

110

, ~u2 =

101

, ~u3 =

011

, ~v1 =

100

, ~v2 =

110

and ~v3 =

111

.

(i) Prove that both U = {~u1, ~u2, ~u3} and V = {~v1, ~v2, ~v3} are bases of R3.

(ii) What are the coordinates of

2−24

with respect to U , and with respect to V ?


(iii) Let the vector ~x have the coordinates (2, 3,−1) with respect to U . What are the coordinates withrespect to the standard basis, and with respect to V ?

(iv) The vector ~x has coordinates (α1, α2, α3) with respect U . What are the coordinates with respect toV ?

Problem 5.5

Let ~v1 =

101

, ~v2 =

110

and ~v3 =

011

.

(i) Find a matrix A with A~v1 = ~v2, A~v2 = ~v3 and A~v3 = ~v1.

(ii) What is the determinant of A? What is the fastest way to find A−1 and A3?

Problem 5.6

Let P1 = (1, 1), P2 = (1,−1), P3 = (2, 1), Q1 = (−1, 4), Q2 = (3, 2) and Q3 = (0, 7).

(i) Is there a linear map which takes Pi to Qi for i = 1, 2, 3?

(ii) Is there a linear map which takes P1 to Q1, P2 to Q3 and P3 to Q2?

Problem 5.7

Let g ∈ R2 be the straight line y =

3

4x+ 2.

Calculate the reflection of (0, 3) about g.

Hint: first describe the reflection at y =3

4x as a linear map with a matrix, then use homogeneous coordinates.

Problem 5.8

Find matrices A1, . . . , A5 representing the following linear maps Li : R3 → R

3:

(i) L1 is the rotation by 3π4 around the z-axis.

(ii) L2 is the rotation by 3π4 around the x-axis.

(iii) L3 is the orthogonal projection onto the x-y-plane.

(iv) L4 is the reflection across the x-z-plane.

(v) L5 is the orthogonal projection onto the straight line ~x = t

122

, t ∈ R.

Problem 5.9

Which vector ~y ∈ R6 with ~y = (m,m,m,m,m,m)⊤, m ∈ R is nearest to

~x = (0, 1, 1, 4, 4, 8)⊤ if you use the norm

(i) || . ||1 (ii) || . ||2 (iii) || . ||∞ ?

Problem 5.10

Let A be a matrix with eigenvalues λ1 = 1 and λ2 = 4 and eigenvectors ~v1 =

[23

]and ~v2 =

[12

].

Find A.

Problem 5.11

General form of reflections

A matrix S defines a reflection iff the following properties hold:


• S = S−1 If you do the reflection twice, everything is unchanged.

• S−1 = S⊤The matrix is orthogonal, so lengths and angles are not altered.

(i) What are the eigenvalues of S?Hint: You may use Jordan decomposition.

(ii) Find all 2 × 2 matrices that define reflexions. Show that there are exactly two such matrices withdeterminant 1. Describe all possible reflections by finding the missing values in

S =

[cosα ∗∗ ∗

].

(iii) Let ‖~a‖ = 1 and P be the hyperplane< ~x,~a >= 0. For~x ∈ R

n, the projection onto the straight line throughzero and ~a is given by < ~x,~a > ~a. (Why?) Thereforethe reflexion of ~x at the plane P is given byS~x = x− 2 < ~x,~a > ~a.

a) Find the matrix representation of S.

b) Show that the matrix has the propertiesS = S−1 = S⊤.

c) In the R2 case, each vector of norm 1 has the

form ~a =

[cos βsin β

]for some value β ∈ [0, 2π).

What is the connection between β and α in (i)?

~x< ~x,~a > ~a

−2 < ~x,~a > ~a

P

~0

S~x = ~x− 2 < ~x,~a > ~a

p

~a

Problem 5.12

The linear map L : R2 → R2 is given by L

[31

]=

[22

]and L

[52

]=

[−13

].

What is L

[13

]?

Problem 5.13

Find matrices A1, . . . , A5 representing the following linear maps Li : R3 → R

3:

(i) L1 is the rotation by 3π4 around the z-axis.

(ii) L2 is the rotation by 3π4 around the x-axis.

(iii) L3 is the orthogonal projection onto the x-y-plane.

(iv) L4 is the reflection across the x-z-plane.

(v) L5 is the orthogonal projection onto the straight line ~x = t

122

, t ∈ R.

Problem 5.14

Find all eigenvalues and -vectors of S =

[cosα sinαsinα − cosα

]and interpret the result geometrically.

Problem 5.15

Let V be the vector space of all polynomials of degree at most three.

T : V → V is defined by (Tp)(x) = p(x)− (x− 1)p′(x).

(i) Show that T is linear.

(ii) Find the matrix of T with respect to the basis 1, x, x2 and x3.


(iii) What is {p ∈ V | Tp = 0}?

Problem 5.16

Let Dφ =

[cosφ − sinφsinφ cosφ

]and Dϕ =

[cosϕ − sinϕsinϕ cosϕ

].

Show Dϕ (Dϕ~x) = Dϕ+φ~x, and try to interpret the result geometrically.


6 Scalar product

Problem 6.1

Let A =

[2 4 4 21 2 2 2

], and U = {~x |A~x = ~0}.

Find an orthonormal basis of U .

Problem 6.2

Let V = C([−1, 1]). A scalar product on V is defined by < f, g >=∫ 1−1 f(t)g(t) dt.

Apply the Gram-Schmidt procedure to the set f1(t) = 1, f2(t) = t and f3(t) = t2.

Problem 6.3

Let a < b and I = [a, b] be an interval. For f, g ∈ C(I) (the space of continous real-valued functions definedon I), the mapping

< f, g > 7→∫

I

f(t)g(t) dt defines a scalar product.

Apply the Gram-Schmidt orthonormalisation process to f1(t) = 1, f2(t) = t, and f3(t) = t2 in the cases

(i) I = [−1, 1]

(ii) I = [0, 1]

Problem 6.4

Let ~v1 =

11−11

, ~v2 =

2002

, ~v3 =

1113

and ~w =

0123

.

Find the orthogonal projection of ~w onto the span of ~v1, ~v2 and ~v3.

Problem 6.5

Let ~u1 =

1013

, ~u2 =

2132

, ~u3 =

2803

and ~z =

06−45

.

(i) Apply the Gram-Schmidt procedure to ~u1, ~u2 , ~u3.

(ii) Find the orthogonal projection of ~z onto the subspace spanned by ~u1 and ~u2.

Problem 6.6

Let U be the subspace of R3 spanned by

13−2

and

22−3

.

What is the projection of

553

onto U?

Problem 6.7

Let

~v1 =

2102

, ~v2 =

1113

and ~v3 =

−4−311

.


(i) Apply the Gram-Schmidt orthogonalisation process to ~v1, ~v2 and ~v3.

(ii) Complete the result of (i) to an ONB of R4.


7 Eigenvalues and Jordan-decomposition

Problem 7.1

Find the Jordan decomposition of A =

3 2 −1−1 0 1−1 1 0

.

Problem 7.2

Find the Jordan decomposition of

A1 =

−2 1 −2 40 −2 8 −140 0 2 10 0 0 2

and A2 =

2 1 0 −10 2 1 −20 0 2 10 0 0 2

Problem 7.3

Let A =

1 −3 −1−3 3 −3−1 −3 1

. Show that the eigenvectors belonging to different eigenvalues are orthogonal. Use

this to find an orthonormal basis of R3 consisting of eigenvectors of A and write

101

as a linear combination

of eigenvectors of A.

Hint: the eigenvalues are 2, −3 and 6

Problem 7.4

Let A =

0 1 11 0 11 1 0

.

(i) Find an orthonormal basis of R3 consisting of eigenvectors.

(ii) Write

−122

as a linear combination of eigenvectors of A.

Problem 7.5

Find all eigenvalues and corresponding eigenvectors of

A =

0 1 1−1 0 11 1 0

and B =

0 1 11 0 11 1 0

Problem 7.6

(i) Prove the following:If A is a real matrix and ~x an eigenvector of the eigenvalue λ, then the vector ~x (the vector consistingof the complex conjugates of the components of ~x) is an eigenvector of the eigenvalue λ.

(ii) Find all (real or complex) eigenvalues and -vectors of A =

1 1 −2−1 1 22 −2 1

.


Problem 7.7

Let A =

[3 12 2

].

(i) Compute a decomposition A = PJP−1, and use this to compute A5.

(ii) Find a matrix B with B2 = A. How many such matrices are there?

Problem 7.8

A =

−4 −3 64 4 −4

−4 −2 6

has the characteristic polynomial p(λ) = −(λ− 2)3.

Write A in the form A = V JV −1 with a Jordan-block matrix J .

Problem 7.9

Construct a matrix A having the eigenvectors ~v1 =

−123

, ~v2 =

403

and ~v3 =

−135

with eigenvalues

λ1 = λ2 = 2 and λ3 = 1.

Problem 7.10

A1 =

5 −3 −32 0 −22 −2 0

A2 =

5 −1 −52 1 −32 −1 −1

A3 =

1 1 1−1 3 10 0 2

(i) The characteristic polynomial of A1 is p(λ) = −(λ− 2)2(λ− 1). Determine the Jordan decomposition.

(ii) The characteristic polynomial of A2 is p(λ) = −(λ− 2)2(λ− 1). Determine the Jordan decomposition.

(iii) The characteristic polynomial of A3 is p(λ) = −(λ− 2)3. Determine the Jordan decomposition.

Problem 7.11

Let J = D +N be a n× n-matrix, D = λEn and N =

0 1 0 · · · 00 0 1 · · · 0...

. . .. . .

. . ....

0 0 0. . . 1

0 0 0 · · · 0

(i) Compute all positive powers of N .

(ii) Compute (D +N)k. (Use the binomial formula)

(iii) For λ 6= 0 compute (D +N)−1

Problem 7.12

The matrix A has the characteristic polynomial p(λ) = (λ− 3)7(λ+ 4)3.

(i) Assume dimker(A− 3E) = 3, dimker(A− 3E)2 = 6 and dimker(A+ 4E) = 1.

What is the Jordan structure of A?

(ii) Assume dimker(A− 3E) = 3, dimker(A− 3E)2 = 5 and dimker(A+ 4E) = 2.

Calculate all possible Jordan structures of A.


Problem 7.13

The matrix A has the characteristic polynomial p(λ) = (λ− 3)7(λ+ 4)3.

(i) Assume dimker(A− 3E) = 3, dimker(A− 3E)2 = 6 and dimker(A+ 4E) = 1.

What is the Jordan structure of A?

(ii) Assume dimker(A− 3E) = 3, dimker(A− 3E)2 = 5 and dimker(A+ 4E) = 2.

Calculate all possible Jordan structures of A.

Problem 7.14

Let A =

2 4 4 64 5 2 04 2 −4 −126 0 −12 −3

.

The characteristic polynomial of A is p(λ) = λ(λ− 9)2(λ+ 18).

Find an ONB of eigenvectors of A.

Problem 7.15

Let ~a,~b ∈ Rn and ~a 6= ~0, ~b 6= ~0.

Find the eigenvalues and -vectors of the n× n-matrix A = ~a~b⊤.

Hints:

(i) What is the dimension of the space spanned by the columns of A, what is the rank of A?

(ii) What is the dimension of the kernel of A, what are the algebraic and geometric multiplicity of λ = 0?

(iii) What is the characteristic polynomial? Find the coefficients of λn and λn−1.

(iv) What vector could be an eigenvector for a non-zero eigenvalue?

(v) What cases have to be distinguished for λ = 0? Find two characteristic 2× 2 examples.

Problem 7.16

Let A =

2 2 2 20 2 2 20 0 2 20 0 0 2

. Find T and J in the Jordan decomposition A = TJT−1.

Problem 7.17

(i) Let x ∈ R, x 6= 0 and N1 =

[x2 10 x2

]. Find all matrices B with B2 = N1.

(ii) Show there is no C with C2 = N2 for N2 =

[0 10 0

].

Hint: Let B =

[a bc d

]and compare B2 with N1.

Problem 7.18

Find the matrices J and P in the Jordan decomposition A = PJP−1 of

(i) A =

−4 −3 64 4 −4−4 −2 6

having the characteristic polynomila p(λ) = −(λ− 2)3.


(ii) A =

4 −1 01 2 14 −3 0

Problem 7.19

Let A =

[1 −11 3

].

Use the Jordan-decomposition of A to find expressions for A2, A3, An and A−1.


8 Quadratic forms

Problem 8.1

Let q1(x, y, z) = x2 + 4xy + 2xz + 6y2 − 4yz + 10z2,q2(x, y, z) = xy + xz + yz and q3(x, y, z) = −x2 − y2 − z2 + 2α(x+ y + z)2.

Are q1 or q2 definite? Are there values of α so that q3 is definite?

Problem 8.2

Find values x1, y1, z1 and x2, y2, z2 so that x2 + 4xy − 2xz + 5y2 − 2yz + z2 is positive for x = x1, y = y1and z = z1 and negative for x = x2, y = y2 and z = z2.

Problem 8.3

Let A =

1 2 22 s 32 3 s

.

Depending on s, decide whether A is definite or not.


9 Numerics of equation systems

Problem 9.1

Let A =

[3 44 5

].

Use the infinity-norm to solve:

(i) What is the condition-number of A?

(ii) Find vectors ~b and δ~b so that the solutions A~x = ~b and A(~x+ δ~x) = ~b+ δ~b fulfill‖δ~x‖‖~x‖ ≥ 8

‖δ~b‖‖~b‖

.

Problem 9.2

Let A =

1 −2 −1−5 6 02 1 4

.

Calculate ‖A‖∞ and ‖A‖1, and find vectors ~x and ~y with ‖~x‖1 = 1 and ‖A~x‖1 = ‖A‖1 and with ‖~y‖∞ = 1and ‖A~y‖∞ = ‖A‖∞.

Problem 9.3

Let A =

[5.7572 −2.8285−8.4857 4.3428

]. Then A−1 =

[4.3428 2.82858.4857 5.7572

].

(i) Calculate the condition-number of A for the ‖.‖∞ and ‖.‖1-norms.

(ii) Find the solution of A~x =

[55−83

].

(iii) Use a theorem from the lectures to find an estimate for the relative error that can occur if you solve

the perturbed system A(~x+∆~x) =

[55.1−82.8

](use the ‖.‖∞-norm.)

(iv) Compare this with the solution of A~x =

[55.1−82.8

].

Problem 9.4

Let ~x be the solution of

[10 911 10

]~x =

[11

].

Calculate ~x and give an estimate for the solution of

[10 9

11.0001 10

]~y =

[11

]. Compare this with the solution

of the perturbed system.


10 Interpolation

Problem 10.1

Find the interpolating polynomial for the values

xi −2 0 1 2

yi 2 0 1 2

Problem 10.2

Write down the Lagrange polynomials L0, L1 and L2 of degree 2 with respect to the points x0 = −1, x1 = 0and x2 = 1.

What are the coefficients of p(x) = 2x2 + 3x− 1 with respect to {Li}?


11 Best-fit problems

Problem 11.1

Compute the best-fit-polynomial of degree 2 for

x −2 −1 0 1 2

y f(−2) f(−1) f(0) f(1) f(2)

where f(x) = cosπx

2.

Problem 11.2

Find the best-fit-straight-line and the coefficient of correlation for the data

xi −1 0 1 2 3

yi −2 0 −1 3 0

Problem 11.3

Let n > 1 and (xk, yk) be points of R2. For simplicity assume that the mean values x = 1n

n∑k=1

xk and

y =1

n

n∑

k=1

yk fulfill x = y = 0. Let at least one x-value and at least one y-value be different from zero.

(i) Calculate the best-fit-straight-line y = vx+ y0 to the given points.

(ii) Express the error F =n∑

k=1

(yk − (vxk + y0))2 with aid of the abbrevations

SX =

√√√√ 1

n− 1

n∑

k=1

x2k, SY =

√√√√ 1

n− 1

n∑

k=1

y2k and SXY =1

n− 1

n∑

k=1

xkyk

(iii) Prove that the coefficient of correlation r =SXY

SXSY

satisfies the inequality −1 ≤ r ≤ 1, and that |r| = 1

is equivalent to F = 0; i.e. that all point lie on a straight line.

Remark. If the conditions x = y = 0 do not hold, the coordinate system may be shifted so that (x, y)becomes the origin.

This results in the formulas

SX =

√√√√ 1

n− 1

n∑

k=1

(xk − x)2, SY =

√√√√ 1

n− 1

n∑

k=1

(yk − y)2 and SXY =1

n− 1

n∑

k=1

(xk − x)(yk − y)

Problem 11.4

Compute the best-fit-straight-line and the coefficient of correlation for

x −2 −1 0 1 2

y −4 −2 1 3 2and

x −5 −3 1 2 5 6

y 0 −1 3 2 4 4


12 Normed spaces

Problem 12.1

Let A =

1 40 −62 −2

.

Find ‖A‖ and a vector ~v with ‖~v‖ = 1 and ‖A~v‖ = ‖A‖ for each of the three (vector and matrix) norms‖.‖1, ‖.‖2 and ‖.‖∞.

Problem 12.2

Let A =

1 −2 2 6−2 5 0 52 −1 1 4

(i) Find ‖A‖1 and a vector ~b with ‖~b‖1 = 1 and ‖A~b‖1 = ‖A‖1.(ii) Solve (i) for the ‖.‖∞-norm.

Problem 12.3

Let A =

0 1 −3−1 0 23 −3 2

.

Find the condition number of A using (i) ‖.‖1 and (ii) ‖.‖∞.

Find a vector ~x 6= ~0 with ‖A~x‖1 = ‖A‖1‖~x‖1 .

Problem 12.4

Let A be an invertible n× n-matrix.

Prove that the condition number of A in the 2-norm is the quotient of the first and last singular values ofA.

Problem 12.5

Let ‖.‖ be the 2-norm, A =

[9 00 1

]and B =

[5 4−4 −5

].

(i) Find the norms of A and A−1, and the condition number of A.

(ii) Find a vector ~v1 with ‖A~v1‖ = ‖A‖‖~v1‖(iii) Find a vector ~v2 with ‖A−1~v2‖ = ‖A−1‖‖~v2‖(iv) Find vectors ~b and ~δb, so that the solutions ~x of A~x = ~b and ~δx of A(~x+ ~δx) = ~b+ ~δb fulfill

‖ ~δx‖‖~x‖ = cond(A)

‖ ~δb‖‖~b‖

(v) Compute the SVD of B.

(vi) Perform steps (i) to (iv) for B.


13 Matrix decompositions

Problem 13.1

Let A =

2 0 2 0−2 −1 −1 24 −1 8 32 0 5 3

and ~b =

432419

.

Find the LU -decomposition of A (without pivoting) and use this to solve A~x = ~b.

Problem 13.2

Let A =

−1 5 −12 −4 21 −3 0

and ~b =

60−5

. Apply LU -decomposition with pivoting to decompose A as

A = PLU with a permutation matrix P , L a lower and U an upper triangular matrix.

Use this to solve A~x = ~b.

Problem 13.3

Solve

1 2 3 4−2 −8 −3 −63 14 5 8−4 −20 −7 −8

~x =

010−3

by constructing an LU-Decomposition (without pivoting).

Let A :=

[2 6−1 −3

]and ~b =

[23

].

Find the singular-value decomposition of A, the Moore-Penrose-inverse and the pseudonormal solution ofA~x = ~b.

Problem 13.4

Calculate the Moore-Penrose-inverse of A =

[2 12 1

]and the pseudo-normal solution of A~x = ~b for~b1 =

[1−1

]

and ~b2 =

[11

].

Problem 13.5

Calculate the singular-value decompositions of A =

1 01 10 −22 1

and of A⊤.

Problem 13.6

Calculate the Moore-Penrose-inverse of A =

1 1 11 1 11 1 1

.

Problem 13.7

Find the solution of

1 2 0 02 1 2 00 2 1 20 0 2 1

~x =

−13−31

. Calculate the LU-decomposition with pivoting.


Problem 13.8

Let A =

2 4 41 2 44 2 4

and ~b =

866

.

Find the solution of A~x = ~b with the short form of the LU-decomposition (with pivoting).

Problem 13.9

Let A =

[34

].

(i) Find the Moore-Penrose-inverses of A and A⊤.

(ii) Find the pseudo-normal-solutions of A~x =

[20

]and A⊤~y = 2.

(iii) Explain the results geometrically.

Problem 13.10

Let A be the 4× 4 tridiagonal matrix A =

2 −1 0 0−1 2 −1 00 −1 2 −10 0 −1 2

Find the LU -decomposition of A with

L =

1 0 0 0l21 1 0 0l31 l32 1 0l41 l42 l43 1

and R =

r11 r12 r13 r140 r22 r23 r240 0 r33 r340 0 0 r44

.

Multiply L and R and find the entries recursively by evaluating the product line by line.

Problem 13.11


1 −1 02 4 10 5 1

~x =

−11313

with aid of LU-decompositions both with and without pivoting.

Problem 13.12

Let A =

1 1−3 −14 01 5

and ~b =

04−22

.

(i) Find the short-form singular-value decomposition of A (you may use the ∗-symbol).

(ii) Find the pseudo-normal solution of A~x = ~b.

Problem 13.13

Let A =

2 1 0 −14 1 2 −20 2 −1 −14 2 −3 1

and ~b =

27−2−2

.

(i) Find the LU -decomposition of A (without pivoting).

(ii) Use (i) to find the solution of A~x = ~b


Problem 13.14

Let A =

0 −20 −143 27 −44 11 −2

.

Find a QR-decomposition of A using

(i) Givens rotations

(ii) Housholder reflexions

(iii) the Gram-Schmidt-procedure

Problem 13.15

Use Housholder reflections to find the Q-R-decomposition of A =

0 6 71 5 −50 8 11

.

Problem 13.16

Write matlab procedures

(i) Start with A = QR = EA

function [Q_new,R_new]=Givens(Q,R,r,c)

Givens shall perform a Givens rotation that sets the element in row r and column c to zero and returnsa pair Qnew, Rnew with QnewRnew = QR, so that a repeated application of this should yield theQR-decomposition of A.

(ii) function [Q_new,R_new]=Householder(Q,R,c)

Householder shall perform a Householder transformation that eliminates the lower part of column c.

If e.g. A is a 4× 4-matrix, the sequence of commands

Q=eye(4)

R=A

[Q,R]=Householder(Q,R,1)



will return a QR-decomposition of A.

Problem 13.17

Use the ‘simple case’ LU-decomposition to solve

1 2 3 4−2 −8 −3 −63 14 5 8−4 −20 −7 −8

~x =

010−3

Problem 13.18

Calculate the singular-value decomposition of A =

12 04 33 −4

Problem 13.19

Let A =

[1 −22 2

]. Calculate the singular-value decomposition of A, A⊤and A−1.


Problem 13.20

Let A = ~x be a matrix with only one column. Find the singular-value decomposition of A.

Problem 13.21

Use LU -decomposition to find the solution of

−u + 2v + 2x = 14u − 6v − w − 7x = 0u + 2x = −1

2v + 3x = 1

Don’t forget to verify your result!


14 Fundamentals of differential equations

Problem 14.1

Find all solutions of

(i) x + 3x = e−2t

(ii) x + x sin t = (sin t)3


15 Linear differential equations with constant coefficients

Problem 15.1

Find the general solutions of

(i) x − 2x + x = 3t2 − 2 sin t

(ii) x + 2x + 2x = et

(iii) x + 2x + x = e−t.

Problem 15.2

Find the general solution in the following cases. In (i), also find the solution with x(0) = x(0) = 0.

(i) x− 3x+ 2x = 2et

(ii) x− 3x+ 2x = 2e−t

(iii) x− 2x+ x = tet

(iv) x− 2x+ x = t2

Problem 15.3

Find a real fundamental system for each of the following problems.

(i) x(3) − 6x+ 12x− 8x = 0

(iii) x(4) + 4x(3) + 8x+ 8x+ 4x = 0

(ii) x(3) + 2x− 3x− 10x = 0

(iv) x(4) − 2x− 8x = 0

Hint: the characteristic polynomial in (iii) is the square of a polynomial of degree two.

Problem 15.4

(i) Find all solutions ofx(3) − 2x+ x = 1 + sin 2t

(ii) Let a0, a1, a2, a3 ∈ R and assume that x1 = tet and x2 = cos t are solutions of the differential equation.

x(4) + a3x(3) + a2x+ a1x+ a0x = 0.

Find a fundamental system and the coeffizients a0 ... a3.

Problem 15.5

Let x(3) − x+ 4x− 4x = f(t). Find all solutions of the homogeneous equation and a particular solution for

(i) f(t) = 4t+ 4

(ii) f(t) = 10et

(iii) f(t) = 4e2t

Problem 15.6


x− x =2

1 + et.

Problem 15.7

Find the general solution of

x(3) − x+ 4x− 4x = −10e−t, x(0) = 4, x(0) = 0, x(0) = −1.


16 Differential equation systems

Problem 16.1

Let x =

1t

t2 0

0 −2t

1t2

0 0 1t

x.

Find a fundamental system and the general solution of the differential equation.

Problem 16.2

Let Y (t) =

cos t sin t

− sin t cos t

, b(t) =

et

e−t

.

(i) Find a differential equation x = Ax so that Y is an fundamental matrix.

(ii) Find all solutions of x = Ax + b.

Problem 16.3


~y ′ =

0 1

−4 4

~y + ex

0

−2

, ~y(0) =

0

0

.

Problem 16.4

Find a fundamental system and the general solution of x =

1t

t2 0

0 −2t

1t2

0 0 1t

x.


17 Differential equation systems with constant coefficients

Problem 17.1

Find the solutions of

(i) x =

2 4

1 2

x.

(ii) x =

3 1

−2 0

x, x(0) =

1

1

.

Problem 17.2

Find a real fundamental system of

x =

0 1 2

−1 0 2

−2 −2 0

x

Problem 17.3

Find fundamental systems of x = Ax for

i) A =

3 1

−2 0

ii) A =

3 1

−4 3

iii) A =

1 −1

4 −3

.

Problem 17.4

Find all solutions of x =

0 −1 2 −2

−1 1 1 0

−2 −1 4 −1

0 0 0 2

x.

The matrix has the eigenvalues λ1 = 1, λ2 = λ3 = λ4 = 2.

Problem 17.5

Find solutions of x =

−3 4

−2 3

+ b(t) for

(i) b(t) =

t

t+ 1

(ii) b(t) = et

1

1


Problem 17.6

Find fundamental systems of x = Ax for

i) A =

3 1

−2 0

ii) A =

3 1

−4 3

iii) A =

0 2 1

−2 0 2

−1 −2 0

.

Calculate a real fundamental system for (iii)

Problem 17.7

Find all solutions of x = Ax for

A1 =

5 −3 −3

2 0 −2

2 −2 0

A2 =

5 −1 −5

2 1 −3

2 −1 −1

A3 =

1 1 1

−1 3 1

0 0 2

Problem 17.8


(i) x =

2 8 0

0 −2 0

0 8 2

x

(ii) x =

4 6

−2 −4

x.


18 Derivatives

Problem 18.1

Let a, b, c, d,An, Bn ∈ R and c > 0.

Show that f1 and f2, defined below are solutions of the diffusion equation ut = c2uxx

(i) f1(x, t) = ax+ b+m∑

n=1

[An sin(nx) +Bn cos(nx)

]e−n2c2t

(ii) f2(x, t) =d√

4c2t+ bexp

(−(x+ a)2

4c2t+ b

).

Problem 18.2

Let U := (0,∞)× (0, π2 ) and Θ : U → (0,∞)× (0,∞) be defined by Θ(

r

ϕ

) =

r cosϕ

r sinϕ

.

(i) Show that Θ is invertible, and calculate the inverse Θ−1.

(ii) Find the Jacobian of Θ.

(iii) Find the jacobian of Θ−1 both directly and with aid of the inverse-function theorem.

Problem 18.3

Let ~f : R3 → R2 be defined by ~f(x, y, z) =

x+ y + z

xy + xz

and ~g : R2 → R

3 by ~g(x, y) =

x2 − 2y

1

2y

.

Calculate the derivative of ~g ◦ ~f both directly and with the chain rule.

Problem 18.4

Let f : R2 → R2 be defined by f(x, y) =

2xy

x2 − y2

and g : R2 → R be defined by

g(x, y) = x2 + y2.

(i) Find the derivative of g ◦ f directly and with the chain rule.

(ii) find the directional derivative of g in (3, 4) in direction of

4

3

.

Problem 18.5

Let f : R2 → R be defined by f(x, y) = y3 − xy + ex − 2.

(i) Show that the equation f(x, y) = 0 has a local resolution (with respect to y) near (0, 1) in the formy = ϕ(x).

(ii) Calculate the Taylor expansion of ϕ up to third order.

(iii) Show that ϕ has a local extremum in x = 0.


Problem 18.6

Let f1, f2 : R2 → R be defined by

f1(x, y) = y3 − y − x2 andf2(x, y) = y3 + y2 − x2.

y

x x

yf2(x, y) = 0

f1(x, y) = 0

(i) Find the points at which the equation f1(x, y) = 0 is not locally resolvable with respect to x or y.

(ii) Where is the equation f2(x, y) = 0 not locally resolvable with respect to x or y?

(iii) In (0, 1) find the second Taylor-polynomial of the resolution y = ϕ(x) of the equation f1(x, y) = 0.

Problem 18.7

For x, y > 0 we define the coordinates r, t by x = r cos2 t and y = r sin2 t.

(i) Show that this definition is meaningful by finding a representation of r and t in terms of x and y.

(ii) Determine

rx ry

tx ty

by differentiating directly.

(iii) Determine

rx ry

tx ty

with aid of the inverse-function theorem.

Problem 18.8

Let f : R2 → R be defined by f(x, y) = x3 + y3 + x2 − y2.

(i) Show that the equation f(x, y) = 0 has a resolution y = ϕ(x) (in a neighbourhood of (0, 1)).

(ii) Find the second Taylor polynomial of the resolution ϕ and show that ϕ has a relative maximum inx = 0.

(iii) Find the third Taylor polynomial of ϕ.

Problem 18.9

Let f : R2 → R2 be defined by f(x, y) = (2xy − 4, x2 − y2 − 3).

From the starting point (x0, y0) = (1, 1) execute two steps of the Newton procedure to find a zero of f .

Problem 18.10

Use the Newton method to find an approximate solution of

2xy − 3 = 0 and x2 − y2 + 4 = 0


19 Taylor series

Problem 19.1

Find, for each of the functions below, the Taylor-series of third order (without remainder) at (xi, yi) :

(i) f1(x, y) = x sin y + sinx, (x1, y1) = (0, 0)

(ii) f2(x, y) = ex+y, (x2, y2) = (0, 0)

(iii) f3(x, y) = (x+ y)3, (x3, y3) = (1, 1)

Problem 19.2

Let f(x) = cos(x+ sin y).

(i) Find the second Taylor-polynomial and the remainder in the point of development (0, 0).

(ii) Use this to compute an approximation of f(0.2, 0.1) and find an estimation of the error.

Problem 19.3

Let f(x, y) =x

1 + y.

(i) Find the second Taylor polynomial of f in (0, 0) and use this to give an approximation of f(0.2, 0.1).

(ii) Find an estimate for the error.

Problem 19.4

Let f(x, y) =√

1 + x+ 2y.

Find the Taylor expansion of first order of f in ~a = (1, 1).

Find an approximation for f(1.2, 1.1), and give an estimate of the error.


20 Extreme values

Problem 20.1

Find all local extreme points of f(x, y) = e−x2−y2(x2 − y2).

Problem 20.2

Let f(x, y) = ex(x2 + 2 + y3 − 3y).

Find all relative extreme points (and their type) of f .


21 Laplace transform

Problem 21.1

Use the Laplace transform to find the solution of

x+ 2x+ x = 3te−t x(0) = 4, x(0) = 2

Problem 21.2

Find the Laplace transform of

(i) f(t) =

0 0 < t < α

1 α ≤ t ≤ β

0 β < t < +∞

(α, β > 0) .

(ii) f(t) =√t .

Problem 21.3


x =

1 −2

2 −3

x +

1

1

, x(0) =

2

1

.

Problem 21.4


x =

0 1

−4 −4

x+ et

0

2

, x(0) =

0

0

using the Laplace transform.

Problem 21.5


x =

−1 1

−1 −1

x+

sin t

cos t

, x(0) =

0

1

using the Laplace transform.

Problem 21.6


d

dtf(t) = cos t− t

2sin t+

∫ t

0f(w) cos(t− w) dw, f(0) = 0.

Problem 21.7

Solve with the Laplace transform

x =

0 1

−4 −4

x+ et

0

2

, x(0) =

0

0

.


Problem 21.8

Find the function f in

(i) f(t) + 3(f ∗ sin)(t) = cos t

(ii) 6f(t) = 6t− t3 +∫ t

0 (t− s)3f(s) ds

Problem 21.9

(i) Find f in the convolution equation f ∗ et = t.

(ii) Use the Laplace transform to find the solution of

x(3) − 3x+ 3x− x = 6et, x(0) = x(0) = x(0) = 0

Problem 21.10

Use the Laplace-transform to find the solution of

x =

4 1

−1 2

x+

e

2t

0

, x(0) =

0

1

Problem 21.11

Calculate the Laplace-Transform of f : R+ → R,

f(t) =

1 n ≤ t < n+ 1/2

0 n+ 1/2 ≤ t < n+ 1, n ∈ N0.

Problem 21.12

Prove the following theorem:

For a periodic function f with period T (f(t) = f(t+ T )) and

f0(t) =

f(t) for 0 ≤ t ≤ T

0 otherwisethe following rule holds:

L [f(t)] (s) =L [f0(t)] (s)

1− e−Ts

Problem 21.13

Find functions with Laplace transform

(i) L[f(t)](s) =1

s(s− 1)3

(ii) L[f(t)](s) =1

s2 + 4s+ 7

(iii) L[f(t)](s) =e−s

s+ 1

(iv) L[f(t)](s) =1− e−s

s


Hint: if neccessary, use partial fraction decomposition.

Problem 21.14

Use the Laplace transform to solve the initial value problem

x+ 5x+ 6x = sinh t, x(0) = 1, x(0) = 4

Problem 21.15

Find the Laplace transform of

(i) f1(t) = cosh kt

(ii) f2(t) = sinh kt

(iii) f3(t) = t2 sin t

(iv) f4(t) = cos2 t.

(v) f(t) =√t .

problems in advanced engineering mathematics

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