EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016

Transcription

1 EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 Answer the questions in the spaces provided on the question sheets. You must show your work to get credit for your answers. There will be eight problems on the actual final exam. Name: Student ID Number:

2 EK102 Practice problems for Final Exam, Page 2 of 15 Spring (15 points) Use the method of elimination to find the complete solution of Ax = b, being sure to state the particular solution and the specific vector(s) in the null space, where A = and b =

3 EK102 Practice problems for Final Exam, Page 3 of 15 Spring (a) (12 points) Use the method of elimination to compute A 1 where A = (b) (3 points) Calculate the determinant of the following matrix B and state why B is not invertible. B =

4 EK102 Practice problems for Final Exam, Page 4 of 15 Spring (15 points) In this problem, you are to consider the matrix A = Find the rank r(a), the column space Col (A), and the null space Nul (A).

5 EK102 Practice problems for Final Exam, Page 5 of 15 Spring (10 points) Let v 1 = 0, v 2 = 1, v 3 = 2 1 number of linearly independent vectors. 2 1, v 4 = Determine the maximum 2

6 EK102 Practice problems for Final Exam, Page 6 of 15 Spring (15 points) In this problem, you are to consider the matrix A = spaces: Col(A), Nul(A). [ ] 1 2. Find the following 4 8

7 EK102 Practice problems for Final Exam, Page 7 of 15 Spring (12 points) A = (a) (3 points) Find all of the eigenvalues of A. [ ] (b) (3 points) For each of the eigenvalues, find an associated eigenvector. (c) (3 points) Sketch both eigenvectors. (d) (3 points) Diagonalize A, being sure to identify the eigenvector matrix P and the eigenvalue matrix D.

8 EK102 Practice problems for Final Exam, Page 8 of 15 Spring (10 points) Determine whether or not b = 1 is a linear combination of the vectors v 1 = , v 2 = 1, and v 3 = 6. Also, if b is a linear combination of these vectors, find that linear combination.

9 EK102 Practice problems for Final Exam, Page 9 of 15 Spring In this problem, you are to consider the matrices A 1 = [ ] 1 2 and A = [ ] (a) (2 points) Find and sketch the images of e 1 and e 2 under the mapping by the matrix A 1. [ ] x1 (b) (2 points) Describe in words how the matrix A 1 acts on general vectors ; recall the words used in Tables 1-4 in Section 1.9. (c) (2 points) Find and sketch the images of e 1 and e 2 under the mapping by the matrix A 2. [ ] x1 (d) (2 points) Describe in words how the matrix A 2 acts on general vectors ; (e) (2 points) Write down the matrix that represents reflection across the line x 2 = x 1. x 2 x 2

10 EK102 Practice problems for Final Exam, Page 10 of 15 Spring Compute A 1 where [ ] 3 9 (a) (3 points) A =, using any method you prefer (b) (7 points) A = 3 1 4, using row reduction

11 EK102 Practice problems for Final Exam, Page 11 of 15 Spring Let v 1 = 3, v 2 = 2, v 3 = 2, v 4 = (a) (8 points) Find the maximum number of linearly independent vectors in H = span{v 1, v 2, v 3, v 4 }. (b) (2 points) Let A be the matrix whose columns are given by the vectors v 1, v 2, v 3, and v 4. Find rank(a).

12 EK102 Practice problems for Final Exam, Page 12 of 15 Spring (10 points) Given A = Determine both Col (A) and Nul (A)

13 EK102 Practice problems for Final Exam, Page 13 of 15 Spring (10 points) Given A = Calculate deta

14 EK102 Practice problems for Final Exam, Page 14 of 15 Spring (10 points) Find all of the eigenvalues and their associated eigenvectors of the matrix [ ] 4 1 A =. 3 6

15 EK102 Practice problems for Final Exam, Page 15 of 15 Spring (10 points) Find all of the eigenvalues and their associated eigenvectors of the matrix [ ] 3 1 A =. 2 5