Exam in Linear Algebra. January 8th, 2013,

Transcription

1 Exam in Linear Algebra First Year at The TEK-NAT Faculty and Health Faculty January 8th,, 9- It is allowed to use books, notes, photocopies etc It is not allowed to use any electronic devices such as pocket calculators, mobile phones or computers The listed percentages specify by which weight the individual exercises influence the total examination This exam set has two independent parts Part I contains regular exercises Here it is important that you explain the idea behind the solution, and that you provide relevant intermediate results Part II is multiple choice exercises The answers for Part II must be given on these sheets Remember to write your full name (including middle names), student number together with the course number on each side of your answers Number each page and write the total number of pages on the front page of the answers Good luck! NAME: STUDENT NUMBER: COURSE: AAU-Cph (Iver Ottosen) AAU-Esbjerg (Olav Geil, Torben Tvedebrink, Leif K Jørgensen) Page of 8

2 Part I ( regular exercises ) Exercise (%) Let [ A = 6 6 ] [, b = ] and c = [ ] Reduce A to its reduced row echelon form Solve the equation Ax = b or argue that it does not have any solution Solve the equation Ax = c or argue that it does not have any solution Exercise (6%) Let A = [ 5 6 ] [, B = ], c = and d = [ ] Determine for each of the following situations if the expression makes sense Evaluate the expressions that make sense For the expressions that do not make sense, write the expression is not defined (Ac) + d AB BA c T A Exercise (5%) Let A = Determine A Page of 8

3 Exercise (8%) Let T : R R be a linear transformation given by x x T x = x x x Determine the standard matrix A for T Determine the standard matrix for the invers linear transformation Exercise 5 (%) In this exercise we are concerned with a linear transformation T : R R Consider the basis {[ ] [ ]} B =, The matrix representation of T with respect to B (also called the B-matrix of T) is given by [ ] [T] B = Find T ([ ]) and T ([ ]) Determine the standard matrix A for T Which geometric operation does T correspond to? Page of 8

4 Exercise 6 (%) Let A = Find the eigenvalues of A Determine for each of the above eigenvalues a basis for the corresponding eigenspace Is A diagonalizable? (justify your answer) Give an argument that A 57 = A Exercise 7 (%) Let W = Span, 5 5 Give an argument that and u =, Determine w in W and z in W such that u = w + z Determine the orthogonal projection matrix P W Find a basis for the orthogonal complement W 5 5 is an orthonormal basis for W Page of 8

5 Exercise 8 (%) The subspace W of R 5 has a basis,, Determine an orthogonal basis for W by using the Gram-Schmidt process Is the orthogonal basis that you found above an orthonormal basis? (justify your answer) Page 5 of 8

6 Exercise 9 (5%) Part II ( multiple choice exercises) The following informations are given A is a symmetric matrix The vector u is an eigenvector for A with eigenvalue equal to Furthermore, the vector v is an eigenvector for A with eigenvalue equal to Exactly one of the following statements is correct Tick off the correct answer u v = u v = u v = Exercise (6%) A is an n n matrix such that det ( A ) = 7 For each of the following two questions tick off the correct answer det(a) = det(a T A ) = Page 6 of 8

7 Exercise (%) Consider u =, u =, u =, u = Let A = [u u u u ] and W = Span{u, u, u, u } Tick off correct statements among the ten statements below (Only the statements that you have ticked off will contribute to your mark Among the statements you have ticked off every incorrect statement will neutralize a correct statement So if you ticked off 5 statements of which are correct and is wrong, you will receive credits for -= correct answers If you ticked of statements of which are correct and are incorrect, you will receive credits for -= correct answer (= no credit) You cannot receive a negative number of points Hence, if you ticked off statements of which is correct and are incorrect you will receive credits for correct answer (= no credit) ) The dimension of W is The dimension of W is The dimension of W is A is invertible (regular) det(a) = {u, u, u, u } is a basis for W {u, u } is a basis for W {u, u } is a basis for W {u, u } is a basis for W Span{u, u, u } = W Page 7 of 8

8 Exercise (8%) Let A = and define the function T : R R by T(x) = Ax Answer the following 5 true/false exercises: a T is a linear transformation b T is onto (surjective) c T is one-to-one (injective) d T is invertible e T corresponds to a rotation Page 8 of 8