Exam in Linear Algebra First Year at The Faculty of IT and Design and at the Faculty of Engineering and Science

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1 For at finde den danske version af prøven, begynd i den modsatte ende! Please disregard the Danish version on the back if you participate in this English version of the exam. Exam in Linear Algebra First Year at The Faculty of IT and Design and at the Faculty of Engineering and Science June 2, 207, This test consists of 0 pages and 4 problems. All problems are multiple choice problems. Your answers must be given by marking the relevant boxes on these sheets. It is allowed to use books, notes, xerox copies 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 problems influence the total examination. Your marks concerning the first three problems will be evaluated as follows: Every wrong mark will annul one correct mark. Remember to fill in your full name (including middle names) together with your student number below. Moreover, please mark the team that you participate in. NAME: STUDENT NUMBER: Team ED CBT (Esbjerg) Ulla Tradsborg Team : BIO KEMI KEMT MILT BIOT Nikolaj Hess-Nielsen Team 2: BA EGI FYS NANO Jacob Broe Team BA MK EN KBT (Esbjerg) Ulla Tradsborg Page of 0

2 Problem (6%) Which of the following statements concerning the system of equations x + 2x 2 x 3 = 5 2x x 2 + x 3 = 0 x + 7x 2 4x 3 = 5 are correct? x = 2, x 2 =, x 3 = 5 solves the system. x = 2, x 2 = 2, x 3 = solves the system x = 2, x 2 =, x 3 = 5 is the only solution of the system. Problem 2 (8%) This problem concerns an investigation of the matrix 2 A = 2. 2 Which of the following statements are correct? u = Null(A). v = 0 is an eigenvector for A. 2 w = is an eigenvector for A. v and w are eigenvectors for the same eigenvalue. v and w are linearly dependent. A is invertible (regular). A is diagonalizable. There is a unique diagonal matrix D which is similar to A. Page 2 of 0

3 Problem 3 (8%) A is an m n-matrix, B is a 2 3 matrix, C = AB is a 3 3 matrix.. What are the values of m and n? m = n = 2 m = 2, n = 3 m = 3, n = 2 m = n = 3 2. A is the standard matrix of a linear transformation S : R n R m, B is the standard matrix of a linear transformation T : R 3 R 2, and C is the standard matrix of a linear transformation U : R 3 R 3. Which of the following statements are wrong, for sure? (a) S is onto (surjective) S is one-to-one (injective) (b) T is onto (surjective) T is one-to-one (injective) (c) U is onto (surjective) U is one to-one (injective) Problem 4 (8%) 2 3 This problem concerns the matrix A = and the vectors b = 8, c = and d = Is d contained in the column space Col A? 2. Is b contained in the column space Col A? 3. Is d contained in the null space Null A? 4. Is c contained in the null space Null A? Page 3 of 0

4 Problem 5 (6%) The matrix 3 0 A = is the standard matrix for a linear transformation T : R n R m.. What is the value of n? What is the value of m? What is the rank of A? What is the dimension of the null space of A? Is T onto (surjective)? 6. Is T one-to-one (injective)? Problem 6 (4%) Which of the following vectors agrees with the product of the matrix 3 [ ] A = and the vector b =? 4 2 [ ] 6 5 none of them Page 4 of 0

5 Problem 7 (6%) This problem is about m m matrices A and B, their products C = BA and D = AB and some of their eigenvectors: v is an eigenvector for A with corresponding eigenvalue 2 and w = Av is an eigenvector for B with corresponding eigenvalue 4. Is it always true that. v is an eigenvector for C? In the affirmative case, what is the corresponding eigenvalue? v is an eigenvector for D? In the affirmative case, what is the corresponding eigenvalue? w is an eigenvector for B 2 = BB? In the affirmative case, what is the corresponding eigenvalue? Problem 8 (6 %) 4 3 The three vectors u = 2, u 2 = and u 3 = 2 form a basis B for R Applying the Gram-Schmidt process to B, yields an orthogonal basis consisting of vectors v, v 2 and v 3. Which of the following statements are correct?. v = u v = u none is correct 2. v 2 = u 2 v2 = u 2 u none is correct 3. v 3 = u 3 v3 = u 3 u 2 v3 = u 3 u u 2 none is correct Page 5 of 0

6 Problem 9 (2 %) This problem concerns the following four matrices A = Is it correct that [ ] [ 32, B = ] , C = 0 2, D = A and B are inverse to each other? 2. C and D are inverse to each other? 3. A is diagonalizable? 4. B is diagonalizable? 5. C is diagonalizable? 6. D is diagonalizable? Page 6 of 0

7 Problem 0 (8 %) [ ] 3 This problem deals with vectors u = and v = 4 transformation T : R 2 R 2 satisfying: T(u) = u and T(v) = v. [ ] 4 in R 3 2 and with a linear Let A = [a a 2 ] denote the standard matrix for the transformation T; hence a and a 2 denote the column vectors of the matrix A. Which of the following statements are correct?. u and v are perpendicular (orthogonal) to each other. 2. u is an eigenvector for T. 3. u + v is an eigenvector for T. [ ] a = 0.28 [ ] a 2 = A is an orthogonal matrix. 7. T describes a rotation in the plane. 8. T describes a reflection in the plane. Page 7 of 0

8 Problem (0 %) This problem concerns the matrix A = and the vector b = Reducing the augmented matrix [A b] = results in the matrix in reduced echelon form.. Which of the columns of A are pivot columns? all four columns only column 2 exactly columns, 3 and 4 exactly columns, 2 and 4 2. What is the rank of the matrix A? What is the nullity of the matrix A? x 4. Is there a solution x = x 2 x 3 to the equation Ax = b with x 2 = 2? x 4 No In the affirmative case, which of the following alternatives is correct for the remaining coordinates? x =, x 3 = 3, x 4 = 4 x =, x 3 = 3, x 4 = 4 x =, x 3 = x 4 = 0 Page 8 of 0

9 Problem 2 (6%) This problem is an investigation of relationships between the three vectors 0 2 a = 2, b = and c = 3 in R c Observe that the last coordinate c of the vector c is a reel variable. Which of the following statements are correct?. a, b and c are linearly dependent vectors for c = 0 c = 4 c = 4 all real numbers c no real number c 2. a, b and c span R 3 for c = 0 c = 4 c = 4 all real numbers c no real number c Problem 3 (8%) This problem concerns (6 6)-matrices A, B and C = AB. It is known that det A = 2 and det C = 6.. What is the value of det( A)? What is the value of det(2a)? What is the value of det B? What is the value of det(b T A)? Page 9 of 0

10 Problem 4 (4 %) The following commands are entered into MATLAB s Command Window: >> A = [ 2 -; 3 ; 2 - ]; >> b = [3; 4; 2]; >> T = [A b];. What is the size of the matrix T? The system of equations Ax = b has a unique solution x. Which of the following sequence of commands results in a calculation of x? >> R = rref(t); x=r(:,4) >> R = rref(a); x=r(:,4) >> R = rref(t); x=r(:,5) >> R = rref(a); x=r(4,:) >> R = rref(t); x=r(4,:) Page 0 of 0