Exercises In Linear Algebra

Exercises in Linear Algebra Group A B C D

1 1 1 1 1

2 2 2 2 2

3 3 3 3 3

4 4 4 4 1

5 5 5 5 2

6 6 6 1 3

7 7 7 2 1

8 8 8 3 2

9 9 9 4 3

10 11 10 11 1 2 5 1 1 2

12 13 14 12 13 14 3 4 5 2 3 4 3 1 2

15 1 6 5 3

A. System of linear equations Solve the following system of linear equations, where a and b are arbitrary constants. 1. x + 2y + 3z = 3 2x + 3y + z = 2 x + 3y + 2z = 3 x+y+z =1 2. x + 2y + 3z + 3w = 9 2x + 3y + z + 6w = 6 x + 3y + 2z + 9w = 9 3. 2x + y = 3 x + y + 2z = 1 y+z =2 4. x+y+z =5 x − y + az = 3 2x + y + z = b 5. x − 2y + 3z = 11 4x + y − z = 4 2x − y + 3z = 10 6. x+y−z =7 4x − y + 5z = 4 6x + y + 3z = 20 7. x + 2y − 3z = 4 3x − y + 5z = 2 4x + y + (a2 − 14)z = a + 2 1

8. x1 + x2 + x3 = a 2x1 + 3x2 + 4x3 = 0 3x1 + 2x2 + bx3 = 0 9. x1 + x2 + 2x3 − x4 = 2 −x1 + 2x2 + x3 + x4 = 3b −2x1 + x2 − x3 + 2x4 = 0 x1 + 2x2 − x3 + ax4 = 1 10. A system of particles consists of three different types, a- particles, b- particles and d- particles . The particles are changing during each time unit in the following way; 20% of the b- particles and 40% of the d- particles to a- particles 20% of the a- particles and 30% of the d- particles to b- particles 50% of the a- particles and 80% of the b- particles to d- particles At a certain time the distribution of particles is (a,b,d) =(500,250,950) Decide the distribution a time unit before and a time unit later. 11. Solve for α, β, γ sin(α) − cos(β) + 3 tan(γ) = 5 sin(α) + 2 cos(β) − 2 tan(γ) = −3 sin(α) − 3 cos(β) + tan(γ) = 5 12. In an electrical network you have found the following relations 2(i3 − i2 ) + 5(i3 − i1 ) = 24 (i2 − i3 ) + 2i2 + (i2 − i1 ) = 0 5(i1 − i3 ) + 2(i1 − i2 ) + i1 = 6 Determine the currents i1 , i2 and i3 . 13. Three liquids if mixed in volume proportion 1:2:4 get the density 0.9. If the proportion is changed to 2:3:1 the density becomes 0.8. And if the proportion is changed to 3:1:5 the density becomes 1. Determine the densities of the liquids. 14. Determine the constants A,B,C and D in a way that makes the curve y + Ax3 + Bx2 + Cx + D = 0 passes the points (1,1) and (-1,5) moreover get a tangent in (2,-1) and becomes parallell to the line 2y − 20x + 11 = 0.

B. General solution, minimal norm and least-squares solution 1. Find the general solution to the equation system x1 + 5x2 + 7x3 + 4x4 =7 2x1 + 4x2 + 8x3 + 5x4 =2 3x1 + 3x2 + 9x3 + 6x4 = −3 2

2. Consider the system of linear equations   

3x1 + 2x2 + x3 + 9x4 = 4 −4x1 + 2x2 − 6x3 − 12x4 = −3 + 7α   x1 + 4x2 − 3x3 + 3x4 = −2 Determine the values of α such that the equation system has exact solutions, and find the general solution. 3. Consider the equation system Ax = b,

A=

1 −1 2 −1 2 0

!

,b =

8 2

!

a. Find the general solution. b. Find the solution with minimal norm ||x|| = xT x. 4. a) Find the general solution to the equation system x1 + 5x2 + 7x3 + 4x4 = 7 2x1 + 4x2 + 8x3 + 5x4 = 2 b) Find the minimal norm (min kxk) solution. 5. An equation system Ax = y is given with 2 1 1 2

   

A=

3 1 1 2

1 1 1 2

4 1 2 3

−9 −3 −5 −8

a) Determine all possible solutions. b) Determine the solution with minimal norm ||x|| =





  , 

  

y=

17 6 8 14

    

√ xT x.

6. Given the following data x −1 0 1 2 y −1 1 2 0 Find an approximate model using the least-square method that closely fit to data above, when the model is a) y = ax + b b) y = ax2 + bx + c 7. Consider the equation system Ax = y below     

1 2 1 2

1 1 2 2 3 5 0 −2 3

2 4 0 6

   x 

 

=  

1 2 3 4

    

a) Show that there exists no solution! b) Find the null space of A. c) Find all least-squares solutions, i.e. that minimizes eT e where e = Ax − y. 8. Given the matrix    

A=

1 0 1 0

0 1 0 1

1 1 1 −1    1 1  1 −1 

a) Find one y ∈ <4 such that there is no solution x to the equation system Ax = y. b) For the chosen y in a), calculate the least-squares solution. c) Find the null space N (A). d) Find the minimal norm least-squares solution for the chosen y in a). 9. The equation error is 





Calculate the least squares approximation that minimizes 

a)





4  and y =  4   −2

e1  e =  e2   = Ax − y e3

P3

2 k=1 ek ,

when



3 2  A=  −4 2  1 4



3 2 1 9   b) A =  −4 2 −6 −12  1 4 −3 3

C. Eigenvalues and eigenvectors 1. The matrix representation of the linear operator A : <2 → <2 is A=

a b b a

!

a, b ∈ <

,

What are the eigenvalues and corresponding eigenvectors of A? 2. The matrix A has the orthogonal eigenvectors 



2   v1 =  1  , 2





1   v2 =  2  , −2





−2   v3 =  2  1

with eigenvalues λ1 = λ2 = 1 and λ3 = −1, respectively. Calculate A. 3. Consider the operator A: <2 → <2 that makes the transformations y = Ax and z = Ay where x=

1 2

!

,

−1 2

y=

Calculate the eigenvalues and eigenvectors of A. 4

!

,

z=

−11 2

!

1 5

4. Consider the matrix





2 1 1  A=  1 2 1  1 1 4 a) Show that λ1 = 1 is an eigenvalue of A and calculate the corresponding eigenvector v1 . b) What are the other eigenvalues and corresponding eigenvectors of A? 



−4 0 0   5. Calculate eigenvalues and corresponding eigenvectors for the matrix A =  0 −7 1 . 0 −9 −1

D. Change of basis Consider a system of differential equations of the kind dx(t) = Ax(t) dt where x is a state (column) vector and A is a matrix. With a transformation of the state vector as x = M xˆ the system can be written as dˆ x ˆx(t) = Aˆ dt where Aˆ = M −1 AM . Choose M as the ”model matrix”, i.e. M = [e1 , e2 , e3 ], where ei , i = 1, 2, 3 are the eigenvectors of A, and calculate Aˆ for the following problems 

1.



4 1 0   A= 2 3 0  0 0 1

2.

A=

α ω −ω α

5



!

3.



8 2 −2   A =  −2 5 4  −2 4 5