Math 340: Elementary Matrix and Linear Algebra

Transcription

1 University of Wisconsin-Madison Department of Mathematics Syllabus and Instructors' Guide Math 340: Elementary Matrix and Linear Algebra Overview. The audience for this course consists mostly of engineering, science and mathematics students who have completed the three-semester calculus sequence. The majority of these students will have seen mathematics mostly as a collection of problem-solving techniques, and they will have had only a very limited exposure to the deductive aspects of the subject. One of the purposes of this course is to serve as a transition between the primarily problem-oriented calculus sequence and the more theoretical 400 and 500-Ievel courses in mathematics. For this reason, it is important to stress the logical interdependence of the ideas that underlie linear algebra and to give actual proofs of at least some of the major theorems. Also, the students should learn to write simple proofs on their own, and problems requiring proofs and justifications should be assigned as homework and should appear on exams. Furthermore, the students should get at least a taste of mathematical abstraction. They should see, for example, that the notion of a vector space (defined by axioms) arises in many different contexts. Nevertheless, it must be remembered that these students are not very sophisticated, and that the primary emphasis should be on understanding the concepts and on using them to solve problems. In order to help instructors attain an appropriate balance between theory and problem-solving techniques, we have included with this syllabus a set of three old Math 340 exams. Some instructors expect their students to learn to use MATLAB, or other software that allows for machine computation of numerical matrix and linear algebra problems. But it should be remembered that Math 340 is definitely not a course in numerical linear algebra. The goal is to get students to understand the concepts; the ability to compute, either by hand or by machine, is secondary, at best. There is barely enough time in the semester to cover all of the material in this syllabus, and so computation should be deemphasized. (Certainly, no class time should be spent on teaching the use of MATLAB.) Exams. We recommend that there should be two midterm exams: one in the sixth week of the semester and one in the eleventh week (approximately). These exams should be given in evenings (5:30-7:00 PM). This avoids consuming scarce lecture time, and it allows the students a little more time to think about some of the theoretical problems. Textbook. We have selected the book Elementary Linear Algebra by B.Kolman and D. R. Hill (8th Edition) as the "standard" text for 340. Instructors who feel strongly that some other book better meets the goals of the course are, of course, free to choose an alternative text. We request, however, that choices of "nonstandard" texts should be discussed with the course coordinators.

2 Topics and schedule. The principal topics in the course are listed below, with the (very) approximate number of (50 minute) lectures to be devoted to each. These total 41 lectures, and since the typical semester contains about 42 or 43 lectures, this schedule allows for very little extra time. It is important, therefore, that instructors should try not to fall behind. Note that the students find that the course suddenly gets much harder when abstract vector spaces are introduced (in Section 3.2 of Kolman and HiIl.) It is vital, therefore, to keep the pace brisk during the initial weeks, so that time needed for the harder material will not be consumed unnecessarily. (NOTE: Feedback from instructors would be welcome concerning the number of lectures budgeted for each topic, and on any other issues of concern.) The following schedule is based on the chapter divisions in Kolman and Hill. Chapters 1 and 2. Linear Equations and Matrices (9 lectures): Matrix algebra, elementary matrices, row operations, inverses, echelon form, Gauss-Jordan elimination. There seems little point in discussing both Gaussian elimination and Gauss-Jordan elimination; \Me recommend that the latter should be stressed. Skip Section 1.1 since linear systems and their matrices are discussed in Sections 1.3 and 2.1. Also, skip the material on partitioned matrices in 1.5 and probably also section 1.6 since time is tight. Do not skip Section 2.3 since equivalent matrices are used later. Omit the "optional" sections: 1.7, 1.8 and 2.4. Note that Problem 28 of Section 1.4 is misleading, if not actually \Mrong. Chapter 3. Real Vector Spaces (12 lectures): Vector space axioms, subspaces, span and linear independence, basis and dimension, rank of a matrix, coordinate vectors. A brief mention of complex vector spaces might also be appropriate here, and Appendix 8.1 can be assigned for reading. Do not lecture on Section 3.1 as most of this material will be familiar to the students. (Assign it for reading, however.) Defer the discussion of isomorphisms in 3.7 until after linear transformations are discussed in Chapter 5. Do not skip the alternative constructive proof of Theorem 3.8 on page 168 since this technique is used again. Theorem 3.10 on page 174 can be proved more transparently. (The book's proof requires the first proof of Theorem 3.8 and not just the statement of that theorem.) Chapter 5. Linear TYansformations and Matrices (6 lectures): Kernel and range, isomorphisms, matrix of a linear transformation, similarity and change of basis. Skip Sections 5.4 and 5.6. Note that we feel that it is natural to introduce linear transformations right after discussing vector spaces, and so \Me recommend doing Chapter 5 immediately after Chapter 3. \Me suggest deferring Chapter 4 to near the end of the course. Chapter 6. Determinants (4 lectures): Odd and even permutations, computation by row and column operations, cofactor expansions, Cramer's rule, inverses of matrices and nonsingularity via determinants. Omit Section 6.6. Note that the proofs of Theorems 6.1 and 6.2 ín the book involve some handwaving about permutations and their signs. Unfortunately, there probably is not enough time to do these proofs properly. Chapter 7. Eigenvalues and Eigenvectors (3 lectures): Definitions and diagonalizalion. Do only Sections 7.1 and 7.2 at this time. The "remark".on page 415 should probably be stated as a theorem from which Theorem 7.5 follows as a corollary. Omit 7.3 and return to 7.4 after doing Chapter 4.

3 Chapter 4. Inner Product Spaces (5 lectures): Cauchy-Schwarz inequality, angle between vectors, Gram-Schmidt process. Students shouid already be familiar with the facts in two and three dimensions, so spend almost no time on that, but assign Section 4.1 (and perhaps also 4.2) for reading. Also, omit Section 4.6 and go easy on orthogonal complements in Section 4.5. In Example 6 on page 237, some mention should be made of why the zero polynomial is the only polynomial defining the identically zero function on an interval. Note that Theorem 4.I2 on page 263 is not quite correct, and (consequently) its proof is sloppy. Chapter 7, continued. Eigenvalues and Eigenvectors (2 lectures): Symmetric matrices. Do only Section 7.4. The book offers two proofs for Theorêm 7.6 on page 427. We recommend doing only the second proof, which seems much more natural, although it requires thinking about eigenvectors in a complex vector space. Unfortunately, the book's presentation is quite terse and it fails to mention that the eigenvector lives in the compler space of column vectors. Sample exams. The three exams on the following pages were given in the fall semester of AIthough they may seem a little difficult now, it is hard to see why today's students should be any less competent than those of Note that a different text was used then, with a slightly different syllabus, and so the timing and the precise choice of topics reflected on these exams do not exactly match the present schedule. Nevertheless, we feel that these exams ind.icate the level of knowledge and sophistication expected from students in Math 340.

4 6th week exam r t ).2-4 1l 1.LetA:l i, 1; ;l L-r o b -11 (a) Find a Row Reduced Echelon Form matrix that is row equivalent to A. (b) Does there exist a nonzero 4 x 1 matrix X such lhal AX : 0? If "yes", find one; if "fro",, prove it. w*r-falz:2 2. Find all solutions, if there are any: w - r -l a - z :0. 3.LetU: r lo 1l lr 1 o ol' wlr-a-z:0 Does [/-1 exist? If so, find it, if not, prove it. Lr o o -2) r 11 0 lo 3 1 o 1l 11-1.Let :lo o 1-1 llandb:lo 3 lo o o 2 zl 11 1 Lo o o o 1l L2 -I a) Find det(,a + Bt). b) Find the second row of AB. c) Find det(.48). d) Find the cofactor Ca1 for the matrix A. À.I is noú invertible. e) Find all numbers À such that B - f) Show that there exists a 5 x 5 matrix X such that AX : B. 5. Let,4, be an n x n matrix. Show that the following are equivalent: (") A is not invertible. (b) A:UV for some nxn matrices U and V, where V has arow of zeros. HINT: Use determinants to deduce (a) from (b). To deduce (b) from (a), you can take I/ to be in RREF r 0l l r 2)

5 llth $/eek exam. Note: In the semester that this exam was given, inner product spaces \ /ere covered earlier than is suggested in the present syllabus. 1.Lett:l\9 o -1 1l LU, i ; _;l. tet V be the solution space for AX:0,'i'e. V is the set of all 5 x 1 matrices X f.or which AX :0. (u) Find a basis for V. (b) Find the dimension of I/. (.) The matrix given to the right is in V. Express it as a linear combination of the basis you found in part (a). 2. Let,4. be an nxn matrix. Let,S be the set consisting of all n x n matrices X such that AX : X,4. Show that,s is a subspace of the vector space of all n x n matrices. 3. In,R3, the set B : {rr,uz,us} is a basis, where ur : (1,2,3), 1)2 : (0, 1, 1) and o3: (1, 1,0). Let S: {i, j,k} be the standard basis for,r3. (u) Express 'd as a linear combination of B. (b) Find the coordinate vector [z]a. (") Find the coordinate vector [rr]r. (d) Find a matrix? such that Tluls: [r]s for all vectors u e R3. (.) Find the fi,rst column of T-r. (Do not compute the whole matrix.) 4. Let Abe a 3 x 7 matrix. Assume that the rows of A are notlinearly independent and that the first two columns of A are linearly independent. Find the rank of A and explain how you determined it. 5. In R4,letW be the subspace spanned by ut: (1,-1,5,3) and uz: (3,-1,7,11)' Let,Ra have the usual inner product. (u) Find an orthonormal basis for W. is inw. Express it as a linear combination (b) The vector (3,1, -1,13) of the basis you found in (a). 6. Let V be the vector space of all differentiable functions. Suppose that f,, g and h arc in I/ and that their derivatives Í',, g' and h' are linearly independent functions. Show that the four functions t, f, g and h are linearly independent in V, where 1 is the constant function with value 1. ll

6 Final exam. 1. Let Z be a linear operator on -R3. Suppose that the matrix of 7 with respect to the bassb:{[i] til li]]" I i i i] Fnd'([i]) l-s 6-2 -b bl 2. Let A : I r 2 7 o o. Let T : R5 -) -R3 be the linear transformation given 112-r-22) by r@) : AX. (u) Find a basis for ker("). (b) Find the nullity and rank of?. (.) Find a basis for the range of Let Pr be the space of real polynomials of degree at most 1. Suppose that P1 is given the inner product defined by (/, ù: Jo dr. Find an orthonormal basis for P1. 4. Let P3 be the space of real polynomials of degree at most 3. Let,S: {1, r,r2,z3} and B: {1,r*!,,(r+L)2,(r+ 1)3}, so that and B are bases for P3. ^9 (u) Find the transition matrix Q from B to S. (b) Let T be the linear operator on P3 deflned bv f U) : (n ]_1)//, where // is the derivative of /. (For example, T(r2 + 7) : (r + 1)(2r) : 2r2 * 2r.) Find the matrix,4. of 7 with respect to,s. (.) Find the matrix Y of T with respect to B. (d) Write an equation relating the matrices Ç, A and Y. Ir o 2l 5. Let : I O t O I. firr a diagonal matrix D and,an invertible matrix P such that l+ o 3l P-L AP : D. 6. Let r, y and z be linearly independent vectors in some vector space. Prove that the three vectors n I U, r -f z and y * z are linearly independent. Prepared by L M. Isaacs and S. Lempp, May Revised December Revised October