(Practice)Exam in Linear Algebra
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1 (Practice)Exam in Linear Algebra First Year at The Faculties of Engineering and Science and of Health This test has 9 pages and 15 problems. In two-sided print. 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 problems influence the total examination. This exam set has two independent parts. Part I contains essay problems. Here it is important that you explain the idea behind the solution, and that you provide relevant intermediate results. Part II is multiple choice problems. The answers for Part II must be given on these sheets. Remember to write your full name (including middle names) together with your student number on each side of your answers. Number each page. Write the total number of pages and the page number on each page of the answers. NAME: STUDENT NUMBER: Side 1 af 9
2 Part I ( Essay-problems ) Problem 1 (8%). Let Find a basis for the column space of A.. Find a basis for the null space of A. 3. Find a basis for the row space of A. Problem (1%). Let Find the eigenvalues of A.. Find a basis for each of the corresponding eigenspaces. 3. Specify whether A is diagonalisable. If so, find matrices P and D, such that D is diagonal, P is invertible, and PDP 1. Side af 9
3 Part II (Multiple-choice problems) Problem 3 (5%) Let R be the row reduced echelon form of the matrix [ ] Specify the value of R 4 : Problem 4 (10%). Consider the matrix Mark all correct statements below (notice: every incorrect mark cancels a correct one). A is not invertible. The linear transformation induced by A is injectiove (one-to-one). A is in row-echelon form. nullity 1. rank 3. nullity A + rank 6. The number 0 is an eigenvalue of A. A is in reduce row-echelon form. There is a vector b R 4, such that Ax = b is not consistent. det 0. Side 3 af 9
4 Problem 5 (8%) Given two 3 3-matrices A og B. Suppose that det 3 and that B is an orthogonal matrix with det(b) > 0. Answer the following questions: a. Specify det B: -.1 b. Specify det(ab): - -3 c. Specify det( A): -3 / Problem 6 (7%). Answer the following 4 true/false questions: a. Let W be a subspace of R 6 having dimension 4. Then dim(w ) =. b. There exists a surjective (onto) linear transformation T : R R 3. c. Suppose Q is a 4 4 ortogonal matrix. Then Q 5 is an ortogonal matrix. d. A 3 3 matrix A with eigenvalues 1, and 3 is both invertible and diagonalizable. Side 4 af 9
5 Problem 7 (5%) Which of the following statements are true (notice: every incorrect mark cancels a correct one): Any orthonormal set in R n is a basis for R n, n > 1. The vectors in an orthonormal set in R n are linearly independent. The vectors in an orthonormal set in R n span R n. The number of vectors in an orthonormal set in R n is at most n. Problem 8 (5%) Let C be given by C = Then det(c) is: -3 -/5 Side 5 af 9
6 Problem 9 (5%) Let and b = Answer the following true/false questions: i. The vector b is contained in Col(A). ii. The vector b is contained in Nul(A). Problem 10 (5%) The following basis is given b 1 = 1 0 0, b = and b 3 = 0 1 1, for R 3. Denote B = {b 1, b, b 3 } and consider the vector 1 v = 1 Answer the following two questions: i. B is an orthonormal basis R 3. ii. The third coordinate of [v] B is given by: 1 3 Side 6 af 9
7 Problem 11 (8%) The row-echelon reduced form of the matrix is given by R = / 1/ Answer the following 4 questions about A: a. The number of pivots of A is: 5 6 b. The number of free variables in the system of equations Ax = 0 is: 5 6 c. Let T be the linear transformation T : R 6 R d given by T(x) = Ax. The number d is: 5 6 d. The linear transformation T(x) = Ax, x R 6, is surjective (onto). Side 7 af 9
8 Problem 1 (5%) Consider the system of equations x 1 + x 3 = 3 x 1 x x 3 = 1 x 1 + x = 4 This system has (mark only one statement): No solution An infinite number of solutions A uniquely determined solution None of the above statements apply. Problem 13 (5%) Consider the matrix Which of the following statements hold true (mark only one statement): A s columns are linearly dependent det(a) = 1 A is not invertible None of the above statements apply. Side 8 af 9
9 Problem 14 (7%) The number of linearly independent eigenvectors of the matrix is given by: 5 Problem 15 (5%) Consider the matrix product AB, where , B = The value of entry (, 4) in AB, i.e. (AB) 4, is: /1. Side 9 af 9
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