Mathematics for Engineering I

Brandenburg Technical University (BTU) Cottbus

Chair of Mathematics for Engineering Prof. Dr. R. Reemtsen, Dr. F. Kemm

Mathematics for Engineering I

Problem Sheet No. 13, January 28/29, 2008 www.math.tu-cottbus.de/˜kemm/lehre/erm

Class Problems

1. Compute the QR factorization and the determinant of the following matrices

A=

1 2 3 4

, B =





√2 √

2 0

0 3 1

√2 √ 2 2√

2



.

Solve the SLE’s A~x = ~a and B~x =~b for the right-hand sides ~a = (0,2)^T and

~b= (4,3√ 2,4)^T.

2. Check if the following functions and mappings are affine or even linear:

a) f : R→R, f(x) = 4x+ 3 , b) f~ : R→R² , f(x) = (4x, x~ ²)^T,

c) f~ : R² →R² , f(x, y) = (xy, x~ +y)^T , d)

f~ : R² →R² , f(x, y) =~

Im ^x+iy_a+ib Re (a+ib)(x+iy)

with fixed real numbers a and b.

Homework

1. Compute the QR factorization and the determinant of the following matrices:

A=





1 1 −1

1 −1 1

−1 1 1



, B =





−2 −2 −1

−3 −1 −1

1 0 −1



.

Solve the SLE’sA~x=~a,B~x=~band C~x=~cfor the right-hand sides~a= (0,2,4)^T and~b= (5,5,0)^T.

2. Let the following matrices be given:

A= √

3/2 −(1 +√ 3/2)

1/2 √

3

, G=

cos(φ) sin(φ)

−sin(φ) cos(φ)

, φ∈R.

Find a value for φ which makes sure that GA is an upper triangular matrix.

3. Check if the following functions and mappings are affine or even linear:

a) f : R→R, f(x) = e^x−e^−x, b) f : R² →R, f(x, y) = x+y+ 1,

c) f : R² →R, f(x, y) = xy+x+y+ 1 , d) f~ : R² →R² , f(x, y) = (x~ −y+ 1, x+y)^T , 4. Check if the following matrix is orthogonal:

1 2









1 1 1 1

1 1 −1 −1

1 −1 1 −1

1 −1 −1 1







 .

5. Compute the inverse and the determinant of the following matrix:

A=





6 −4 7

−12 5 −12

18 0 22



.

6. Compute the LU factorization of the following matrix:

A=









2 0 −1 0

0 −1 2 1

2 1 −2 −1

4 −3 5 6







 .

Solve the SLEA~x=~bfor the right-hand side a) ~b= (1,0,2,3)^T,

b) ~b= (1,−1,1,−2)^T.