Brandenburg Technical University (BTU) Cottbus
Chair of Mathematics for Engineering Prof. Dr. R. Reemtsen, Dr. F. Kemm
Mathematics for Engineering I
Problem Sheet No. 10, January 7/8, 2007 www.math.tu-cottbus.de/˜kemm/lehre/erm
Class Problems
1. Find a solution of A ~x=~bfor
A=
1 1 1 1 2 1 3 0 1
, ~b= (6,8,6)T.
2. Let the following homogeneous system of linear equations be given:
x1−x2+ 2x3−5x4−3x5 = 0, x1− 1
2x2 +x3−3x4−3x5 = 0, 2x1−x2+ 3x3−9x4−7x5 = 0, x1− 1
2x2+ 2x3−6x4−4x5 = 0, 2x1−3
2x2+ 4x3−11x4−7x5 = 0. a) Perform the Gauss elimination.
b) Find two particular solutions.
c) Give the complete set of solutions for the system of linear equations.
3. Solve the following homogenous system of linear equations:
x1 +x2+ 1x3+x4 = 0 , 2x1+ 2x2+ 5x3+ 2x4 = 0 , 3x1+ 3x2+ 4x3+ 3x4 = 0 , 4x1+ 4x2+ 4x3+ 4x4 = 0 .
4. Let the following system of linear equations be given:
x1−3x2+ 3x3−x4 = 1,
−2x2+x3−x4 = 1.
a) Find two linearly independent special solutions of the corresponding homo- geneous system.
b) Express the solution set of the homogeneous system in terms of these special solutions.
c) Find the solution set of this inhomogeneous system with a particular solution of the inhomogeneous system.
5. Find out whether the following vectors are linearly independent. In case of linear dependency give a basis of the space spanned by the three vectors.
a)
1 1
−1
,
−2
−1 1
,
0 1
−1
.
b) 1
√5
0 2 1
, 1
√6
1
−1 2
, 1
√2
1 1 0
.
6. Let the following vectors be given:
~a=
1 1 1 1
, ~b=
1 2 3 2
, ~c=
1 2 3 3
, d~=
1 2 3 5
.
a) Check whether~a,~b, ~care linearly dependent or independent. In case of depen- dency give a basis of the space spanned by these vectors.
b) Check whether ~a,~b, ~c, ~d are linearly dependent or independent. In case of dependency give a basis of the space spanned by these vectors.
7. Represent the vector~b := (1,3,2)T as a linear combination of the linearly inde- pendent vectors and check your result:
~
a1 := (1,1,0)T, a~2 := (0,1,−1)T, a~3 := (2,−1,1)T.
Homework
1. Find all solutions of the following system of linear equations:
x1−x2+ 2x3 =−1, 2x1+ 2x3 = 2 , 3x1+x2+ 3x3 = 4 .
2. Find all solutions of the following system of linear equations:
3x1+ 2x2−x3+ 4x4 = 0, 3x1+ 7x2−x3+ 16x4 = 9, 3x1 + 2x2−x3−10x4 = 16, 6x1+ 4x2 −4x3+ 14x4 = 4, 9x1+ 6x2 −3x3+ 12x4 = 12.
3. Solve the following system of linear equations:
−4x1+ 6x2+ 20x3 = 20,
−2x1+ 5x2+ 15x3 = 19,
−7x1 + 12x2+ 40x3 = 43.
4. Let the following homogeneous system of linear equations be given:
2x1−x2+ 1x3−3x4−5x5 = 0, x1− 1
2x2 +x3−3x4−3x5 = 0, 3x1−x2+ 2x3−7x4−9x5 = 0, 2x1− 1
2x2 +x3−4x4−6x5 = 0, 4x1−3
2x2+ 2x3−7x4−11x5 = 0. a) Perform the Gauss elimination.
b) Find two particular solutions.
c) Give the complete set of solutions for the system of linear equations.
5. Let the following system of linear equations be given:
2x1−3x2+ 3x3−x4 = 2 , 3x1−2x2+x3−x4 =−1.
a) Find two linear independent special solutions of the corresponding homoge- neous system.
b) Express the solution set of the system in terms of these special solutions.
c) Find the solution set of this inhomogeneous system with a particular solution of the.
6. Solve the following homogenous system of linear equations:
x1+ 2x2+ 3x3+ 4x4 = 0, 2x1+ 4x2+ 6x3+ 10x4 = 0, 3x1+ 6x2+ 9x3+ 15x4 = 0, 5x1+ 10x2+ 15x3+ 25x4 = 0.
7. Find alla ∈Rwhich make the following vectors linearly independent:
−2 1 1
,
0 1 a
,
−1 0 1
.
8. Let the following vectors be given:
~a=
1 1 1 1
, ~b=
1 2 3 4
, ~c=
1 4 7 10
, d~=
1 4 10 16
, ~e =
1 4 10 20
.
a) Check whether~b,~c ~d, ~e are linearly dependent or independent. In case of de- pendency give a basis of the space spanned by these vectors.
b) Check whether~a,~b, ~c, ~d, ~e are linearly dependent or independent. In case of dependency give a basis of the space spanned by these vectors.
9. Represent the vector~b := (2,1,3)T as a linear combination of the linearly inde- pendent vectors and check your result:
~a1 := (0,1,1)T, ~a2 := (−1,0,1)T, ~a3 := (1,2,−1)T. 10. Solve the following equations:
√x−1−√
x−4 = 1,
√5−x−√
x−1 = √
2x+ 2−√
2−2x ,
|x+ 1|+|x−1|=|x|.
