Syllabus, Math 343 Linear Algebra. Summer 2005

Transcription

1 Syllabus, Math 343 Linear Algebra. Summer 2005 Roger Baker, 282 TMCB, phone extension Welcome to Math 343. We meet only 20 times (see the calendar in this document, which you should keep handy). Nevertheless, we cover all the essentials in the course book (Roger Baker, Linear Algebra, Rinton Press 2001). If you want to know about linear algebra in economics, engineering, computer graphics and other interesting and important applications, browse in David Lay, Linear Algebra and Its Applications in the library. In real world applications, you will use a combination of theory and computer programs. In this course, we just want to learn the basic theory, and it is not hard to do all the required calculation by hand. I shall stick to one nice application (Markov chains). I should emphasize that accurate arithmetic in your calculations is a crucial part of your course. I can t give much credit if you do a calculation wrongly, because you have not done the required task. In most cases there are neat ways to check your work and I will, repeatedly, show you these. Once more, for emphasis: if you want a good grade you have to earn it by checking every calculation you make. It should help that the exams are take-home, and you don t have to race the clock when doing your checking. Regarding exams, you may not discuss them with anyone (your spouse, parent, classmate, etc.). If you do, you are in breach of the honor code. But you can come and talk to me and I can help, in the sense of doing relevant problems with you. My office is in 282 TMCB. I want to go over a good many homework questions in class, so if you hand in homework late, there is no credit if I already did this. You also must have a really good excuse to get permission to hand in homework late. Obviously every class and every homework is important with our compressed schedule. Here are a few more important facts. (i) Homework counts for 30%, the take-home midterm for 35%, and the take-home final for 35% of your grade. Homework questions are out of 10 and all count equally. (ii) I don t grade on a curve. I would be only too pleased to give an A to everyone if your work is good. (iii) You have to get a course total of 90% for an A, 80% for a B and 70% for a C. (iv) Homework is due on Wednesday (covering Monday s lecture) and Monday (covering Wednesday s and Friday s lectures). The complete set of homework questions (with due dates) is attached. If you lose them, go to my faculty web page at where this document can be found. I intent to give our printed solutions once I have gone over homework in class, but it is important to listen to the class explanations of solutions to extract as much information as possible. (v) There are holidays on two Mondays (July 4 and 25). Homework that would normally be due then is due on the 6th or 27th July instead. (vi) I can be found in my office before the lecture (11-12) and after (2-3) on MWF. On Tuesday you can come between 11 and 2 to ask questions. I can provide expert help

2 with homework by doing similar questions. With this service available, you should not have trouble turning in good answers! You can also work in groups on homework, but you must write up your work individually once you have roughed out solutions in a group. You must keep all homework and exams once they are returned, since sometimes I need to take a second look to adjust scores. It is a very good idea to make a final draft of the exams before turning them in. Messy work is unlikely to be accurate or clearly explained. Here are some standard department policies. Preventing Sexual Harassment: BYU s policy against sexual harassment extends not only to employees of the university but to students as well. If you encounter unlawful sexual harassment, gender-based discrimination, or other inappropriate behavior, please talk to your professor; contact the Equal Employment Office at ; or contact the Honor Code Office at Students with Disabilities: BYU is committed to providing reasonable accommodation to qualified persons with disabilities. If you have any disability that may adversely affect your success in this course, please contact the University Accessibility Center at Services deemed appropriate will be coordinated with the student and instructor by that office. Math Summer Calendar Sections covered (1.2 Chapter 1 Section 2) June M W , Homework 1 due F M Homework 2 due W Homework 3 due July F W Homework 4 due F M Homework 5 due W , 4.7 Midterm given out F M Homework 6 due W Midterm due F W Homework 7 due F Aug. M Homework 8 due W 3 7.2, 7.4 Homework 9 due F 5 8.1, 8.3 Final given out M Last class. Homework 10 due F 12 Final due (under door of 282 TMCB). The midterm covers Chapters 1 4 and the final, Chapters 5 8.

3 Homework 1. Due in class Wednesday, 22 June. 1. Find the equation of the straight line through (7, 8) perpendicular to the direction (3, 1). 2. Find the sum of the three vectors (4, 5, 7), (3, 1, 2) and (0, 1, 4). 3. Among the vectors (1, 2, 3), (2, 1, 4), (5, 6, 4), (6, 9, 4), find all the pairs of perpendicular vectors. 4. Find the distance from the point (3, 5) to the straight line 6x 1 + 5x 2 = Among the vectors (7, 2, 3), (1, 5, 1), (4, 9, 4), which pair has the smallest angle between them? 6. Find the point on the line 3x 1 + 2x 2 = 7 obtained by dropping a perpendicular from (4, 2). 7. Find the point that is one-fourth of the way along the line segment starting at (7, 11) and ending at (6, 3).

4 Homework 2. Due in class Monday, 27 June. 1. Find the parametric equation of the straight line through (1, 2, 1) and (4, 6, 3). 2. Find the equation of the plane passing through (3, 7, 1), (0, 1, 2) and (1, 2, 0). 3. Write the plane with equation in parametric form. 2x 1 + 5x 2 x 3 = A normal to the plane with equation 5x 1 + x 2 7x 3 = 10 passes through the point (2, 1, 2). Where does this normal meet the plane? 5. Show that the midpoints of the sides of the quadrilateral in R 2 with consecutive vertices at p, q, r, s always form the vertices of a parallelogram. s r p q 6. Find the projection of (6, 2, 1) on (3, 3, 4). 7. Check that the Cauchy-Schwarz inequality is satisfied by the pair of vectors (7, 1, 9, 1) and (2, 3, 5, 0). 8. Using the formula on p. 32, find the distance from the point (1, 2, 1, 3) to the hyperplane with equation x 1 + x 2 x 3 + x 4 = Write the linear system as a vector equation. x 1 + x 2 x 3 + 2x 4 = 7 x 1 + x 2 + 4x 3 = 1 x 2 + x 3 + 4x 4 = 0

5 10. Find the augmented matrix of the system x 1 + x 2 x 3 + 2x 4 = 9 x 1 x 2 7x 3 + 5x 4 = Write down the coefficient matrix of the system 6x 1 + 5x 2 + 3x 3 = 11 x 1 x 2 7x 3 = Which of the following matrices are in reduced echelon form? Explain [ ] , 0 0 1, , Row reduce the following matrices to reduced echelon form. Show your row operations as in Example 4, p. 43. [ ] , 5 1 7, , Find the pivot columns of the matrices in question 13.

6 Homework 3. Due in class Wednesday, 29 June Solve exercises 4.1, 4.2, 4.3, 4.5, 4.6, 4.7 for Chapter Show that (1, 2, 5) is a solution of the system 3x 1 + 5x 2 4x 3 = 7 (1) 3x 1 2x 2 + 4x 3 = 13 6x 1 + x 2 8x 3 = 32. You may assume that the general solution of the system 3x 1 + 5x 2 4x 3 = 0 (2) 3x 1 2x 2 + 4x 3 = 0 6x 1 + x 2 8x 3 = 0 is x 3 ( 4 3, 0, 1). Without calculation, write down the general solution of the system (1).

7 Homework 4. Due in class Wednesday, 6 July. 1. Let w 1, w 2, w 3 be vectors in R n such that w 3 = 6w 1 9w 2. Write the linear combination a 1 w 1 + a 2 w 2 + a 3 w 3 in the form (i) b 1 w 1 + b 2 w 2 (ii) c 1 w 1 + c 2 w 2 (iii) d 1 w 2 + d 2 w Determine whether the column vectors of the matrix A = are linearly independent. 3. Determine whether the column vectors of the matrix A = are linearly independent. 4. Determine whether the column vectors of the matrix A = are linearly independent. 5. Let v 1, v 2 and v 3 be three nonzero vectors in R 3. Suppose that the angles between them are as follows. Is v 1, v 2, v 3 a linearly independent set? v 1 and v 2 : 10 ; v 2 and v 3 : 20 ; v 1 and v 3 : 40.

8 6. Let v 1 = (5, 1, 0, 0), v 2 = (4, 1, 2, 0), v 3 = (0, 0, 1, 0), v 4 = (2, 1, 5, 6). Show (without much calculation if possible) that v 1, v 2, v 3, v 4 is a linearly independent set. 7. Let V be the subspace of R 4 consisting of the solutions of x 1 + 7x 2 + x 3 + 2x 4 = 0 x 1 x 2 7x 3 14x 4 = 0. Show without calculation that v 1, v 2 is a basis of V, where v 1 = (6, 1, 1, 0), v 2 = (12, 2, 0, 1). 8. Let V be the subspace of R 7 consisting of the solutions of a homogeneous system with coefficient matrix A: a 11 x 1 + a 12 x a 17 x 7 = 0 a 21 x 1 + a 22 x a 27 x 7 = 0 a 31 x 1 + a 32 x a 37 x 7 = 0 Let v 1, v 2, v 3, v 4, v 5 be a basis of V. What can you say about the number of pivot columns of A?

9 Homework 5. Due in class Monday, 11 July. 1. For each of the following matrices A, find the rank and nullity of A. [ ] [ ] ,, 2 6 9, For the matrices in question 1, find a hidden relation between the columns of A (if there is one). 3. Let v 1 = (1, 2, 6, 4), v 2 = ( 1, 3, 5, 7), v 3 = (2, 2, 0, 1). Find the equation of the hyperplane H that contains v 1, v 2 and v 3. Explain why it is true that 4. Let T : R 3 R 4, H = Span{v 1, v 2, v 3 }. T (x 1, x 2, x 3 ) = (x 1 + x 2 + x 3, x 1 x 2 + 3x 3, 7x 1 + x 2, x 2 + x 3 ). Show by a direct calculation that T (x + y) = T x + T y and T (cx) = ct x. Find T a 1, T a 2 where a 1 = (1, 1, 1), a 2 = (3, 2, 4). Find T (a 1 + a 2 ) with as little computation as possible. 5. Find the matrix of the linear mapping T in question Let T be an anticlockwise rotation through 135 of R 2. Find the image T x when x = (3, 2). 7. Let T : R 2 R 2 be a rotation through angle a. Using the matrix formula for T, show that T x = x. 8. If T is a rotation of R 3 about the vertical axis e 3, through angle a, what is T x? Is T linear? 9. Show by a row reduction that the inverse of the linear mapping T (x 1, x 2 ) = (7x 1 x 2, 2x 1 + x 2 ) is S(y 1, y 2 ) = ( y 1 +y 2, 2y 1+7y 2 ) 9 9.

10 Homework 6. Due in class Monday, 18 July. 1. Determine the parity of the following permutations. (i) 1, 2, 4, 3, 6, 5 (ii) 1, 4, 5, 2, 3 (iii) 7, 6, 5, 4, 3, 2, 1 (iv) 1, 3, 5, 7, 2, 4, 6, Let A = Find the inverse of A. Verify that your answer B has AB = I. 3. The matrix A = a 0 c 0 0 d 0 e f is invertible except for certain unlucky choices of a, c, d, e, f. Find out what these choices are, and find the inverse of A if we avoid these values. 4. Find the determinants of the following matrices. [ ] (i) (ii) (iii) Which of these matrices are invertible?

11 1. Find the adjoint of Math Summer 2005 Homework 7. Due in class Wednesday, 27 July. a b a + b A = c d c + d. e f e + f Work out A adj A. Was this answer predictable? 2. Find the adjoint of a 1 A = a 2 a 3 a 4 and work out A adj A. Does your answer fit with Proposition 12 of Chapter 5? 3. Solve the system x 1 + 2x 2 + x 3 = 8 x 1 + 4x 2 + x 3 = 6 6x 1 + 6x 2 x 3 = 29 using Cramer s rule. (The answers are whole numbers.) 4. Find the volume V of the parallelepiped with vertices a, b, c adjacent to 0, where a = (3, 6, 2), b = (1, 3, 5), c = (2, 1, 3). 5. Show that 1 x x 3 det 1 y y 3 = (y x)(z x)(z y)(x + y + z). 1 z z 3 Hint: Use row operations to simplify the determinant. Extract factors from rows where possible. 6. Let b 1, b 2 be real numbers with b b 2 2 = 1. Find the characteristic polynomial of the projection S on page 88. Can you explain why the polynomial does not depend on b 1 and b 2?

12 Homework 8. Due in class Monday, 1 August. For each of the following matrices A, find P and diagonal D such that A = P DP 1. (Or, show A is not diagonalizable.) 1. A = [ ] A = A = Let Explain why lim m Am exists A =

13 Homework 9. Due in class Wednesday, 3 August. 1. Let A be the transition matrix A = Show that there is a steady state vector q that does not depend on q 0, and find q. 2. Let A be the transition matrix A = If the initial probability vector is (c 1, c 2, c 3, c 4 ), find the steady state vector. Hint: This is similar to the argument on pp Notice that in solving question 2, all the eigenvectors for eigenvalues 1 have coordinates that add up to 0. This is true for every transition matrix. Prove this by following the procedure outlined. Let Ax = cx where A is a transition matrix, c 1 and x 0. Write this out as a 11 x a 1n x n = cx 1, a 21 x a 2n x n = cx 2, and so on. Now add up all n equations. After simplifying, you should end up with x 1 + x x n = c(x 1 + x x n ). Now you should be able to complete the argument.

14 Homework 10. Due in class Monday, 8 August. For each of the following linearly independent sets, find an orthogonal basis of the linear span of the set. Aim for the simplest looking basis. Check orthogonality! 1. w 1 = (1, 2, 3), w 2 = ( 1, 2, 3). 2. w 1 = (0, 1, 2, 1), w 2 = (1, 3, 1, 1), w 3 = (1, 0, 2, 2). (You should be able to divide by 9 when constructing v 3.) For each of the following symmetric matrices A, find an orthogonal matrix P such that 3. A = [ ] A = P DP t, D diagonal A = 0 0 5/2. 0 5/ A = Compute z z and z for the following complex numbers a. (i) 3 + 4i (ii) 1 2i (iii) 5 + 6i (iv) i (v) 12 5i 7. Compute w/z for the following pairs of complex numbers: (i) w = 1 + i, z = 2 5i (ii) w = 3i, z = 4 + 7i (iii) w = i, z = 1 i (iv) w = 10, z = 1 + 2i (v) w = i, z = 5 12i