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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.
Topics and schedule.
The principal topics in the course are listed below, with^ the^ (very)^ approximate^ num-
ber 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^ elim- ination; \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.
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^ The- orems 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^ diagonal- izalion. 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.
and return to 7.4 after doing^ Chapter^ 4.
6th week exam. | t 0 1).2 .3^ 1r 1.LetA:l -4^ 1l i, 1; ;l L-r o b (^) -
(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:
- Find all solutions, if there are any: w (^) - r -l^ a (^) - z^ :^. wlr-A-z:
3.LetU: Does [/-1 exist? If so, find it, if^ not,^ prove^ it.
11 0 2 4 -1r 11 0 lo 3 1 o 1l (^11) - .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. e) Find all^ numbers^ À^ such^ that B^ - À.I^ is^ noú^ invertible. f) Show that there exists a 5 x^5 matrix^ X^ such that^ AX^ :^ B.
- 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.
11 0 1 0r lo 1 o 1l lr 1 o ol' Lr o o (^) -2) 0 0 1 0 1
0 0r 0 0l 0 01. 1 0l r 2)
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).
- 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.
- 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.)
- 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.
- 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. (b) The vector (3,1,^ -1,13) is^ inW.^ Express^ it^ as^ a^ linear^ combination of the basis you found in (a).
- 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.
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