Math2Sheet14

University of Leipzig - SS19 10-PHY-BIPMA2 - Mathematics 2 / Vitalii Konarovskyi Problem sheet 14 Tutorials by Ikhwan Khalid < [email protected] > and Mahsa Sayyary < [email protected] > . Solutions will be collected during the lecture on Wednesday July 10. Points for solved exercises have to be included as bonus points for the homework 1. [2 points] Solve the following systems of linear equations:      2 x 1 + 7 x 2 + 3 x 3 + x 4 = 6 , 3 x 1 + 5 x 2 + 2 x 3 + 2 x 4 = 4 , 9 x 1 + 4 x 2 + x 3 + 7 x 4 = 2 . 2. [2 points] Find the fundamental system of solutions of the following system of homogeneous linear equations:            3 x 1 + 2 x 2 + 5 x 3 + 2 x 4 + 7 x 5 = 0 , 6 x 1 + 4 x 2 + 7 x 3 + 4 x 4 + 5 x 5 = 0 , 3 x 1 + 2 x 2 - x 3 + 2 x 4 - 11 x 5 = 0 , 6 x 1 + 4 x 2 + x 3 + 4 x 4 - 13 x 5 = 0 . 3. [2+3 points] Compute the following determinants: - 3 9 3 6 - 5 8 2 7 4 - 5 - 3 - 2 7 - 8 - 4 - 5 , 1 2 3 . . . n - 2 n - 1 n 2 3 4 . . . n - 1 n n 3 4 5 . . . n n n . . . . . . . . . . . . . . . . . . . . . n n n n n n n . 4. [3 points] Compute the rank of matrix   1 λ - 1 2 2 - 1 λ 5 1 10 - 6 1   depending on λ . 5. [3 points] Show that the following systems of vectors form bases in R 3 , find the change of basis matrix and find the coordinates of vector x = (1 , 0 , 1) in both bases. The first system: e 1 = (1 , 1 , 0), e 2 = (0 , 0 , 1), e 3 = ( - 1 , 1 , 0) and the second system: e 0 1 = (1 , 2 , 1), e 0 2 = (2 , 3 , 3), e 0 3 = (3 , 7 , 1). 6. [2 points] Find the dimension and a basis of the linear subspace span { v 1 , v 2 , v 3 , v 4 , v 5 } of R 4 , where v 1 = (1 , 0 , 0 , - 1), v 2 = (2 , 1 , 1 , 0), v 3 = (1 , 1 , 1 , 1), v 4 = (1 , 2 , 3 , 4), v 5 = (0 , 1 , 2 , 3). 7. [2 points] Determine whether the matrix   5 2 - 3 4 5 - 4 6 4 - 4   can be reduced to a diagonal form by going over to a new basis. Find that basis and the corresponding matrix. 1

University of Leipzig - SS19 10-PHY-BIPMA2 - Mathematics 2 / Vitalii Konarovskyi 8. [3 points] Let a linear map T : R 3 → R 3 be defined by T ( x, y, z ) = ( x, x + y, x + y + z ). Find its matrix in the standard basis and also in the basis v 1 = (1 , 0 , 0), v 2 = (2 , 0 , 1), v 3 = (0 , 1 , 1). Check if T is invertible and if yes, find the inverse map T - 1 . Is T self-adjoint? 9. [3 points] Find the orthogonal projection of the vector u = (4 , - 1 , - 3 , 4) onto span { v 1 , v 2 , v 3 } in R 4 with the standard inner product, where v 1 = (1 , 1 , 1 , 1), v 2 = (1 , 2 , 2 , - 1), v 3 = (1 , 0 , 0 , 3). 10. [4 points] Consider the vector space R n [ x ] of all polynomials with real coefficients of degree at most n with the inner product h p, q i = Z 1 - 1 p ( x ) q ( x ) dx. Show the following polynomials p 0 ( x ) = 1 , p k ( x ) = 1 2 k k ! d k dx k ( x 2 - 1) k , k = 1 , . . . , n, form a basis in R n [ x ]. 11. [3 points] Show that A is self-adjoint and compute e A , if A = 3 - i i 3 . 12. [2 points] Reduce to a canonical form the following quadratic form on R 3 Q ( x ) = x 2 1 - 2 x 2 2 + x 2 3 + 2 x 1 x 2 + 4 x 1 x 3 + 2 x 2 x 3 . 13. [2 points] Show that A ∪ B = A ∪ B , where A denotes the closure of a set A . 14. [3 points] Compute the partial derivatives ∂f ∂x (0 , 0) and ∂f ∂y (0 , 0), if f ( x, y ) = 3 √ xy . Is the function f differentiable at (0 , 0)? 15. [3 points] Does ∂ 2 f ∂x∂y (0 , 0) exist, if f ( x, y ) = ( 2 xy x 2 + y 2 , if x 2 + y 2 > 0 , 0 , if x = y = 0? 16. [3 points] Show that the function z = z ( x, y ) defined by the equation F ( x - az, y - bz ) = 0 , where F is a differentiable function of two variables and a, b are some constants, solves the equation a ∂z ∂x + b ∂z ∂y = 1 . 17. [2+2+2 points] Find the general solutions to the following equations: a) y 0 ln | y | + x 2 y = 0; b) xy 0 + (1 + 2 x 2 ) y = x 3 e - x 2 ; c) y 0 = y 2 +2 xy x 2 . 18. [3+3 points] Solve the initial value problems: a) y 0 - xy = xy 3 2 , y (1) = 4; b) y (4) - 16 y = 0, y (0) = y 0 (0) = 2, y 00 (0) = - 2, y 000 (0) = 0. 2