## MATH 332-01 Matrices (Ellis) Fall 2010

 Instructor: Robert Ellis Office: E1 Bldg, Rm 105C Email: Lectures TR 11:25am-12:40pm Perlstein Hall 109 Office Hours MW Walk-in, TR by appointment Office phone 567-5336 Appointments and emailed questions are welcome Textbook Anton, Elementary Linear Algebra, 9th edition, Wiley

Test Dates
Quiz 1: Thursday, October 7 (quiz, key)
Exam 1 (Through 4.1): Thursday October 14 (see Blackboard->Course Documents for key)
Quiz 2: Tuesday, November 16 (quiz, key)
Exam 2 (Chapter 4 and 5.1-5.5): Tuesday, November 23 (see Blackboard->Course Documents for key)
Final Exam: Tuesday, December 7, 10:30am-12:30pm

Sample cover sheet for homework done in groups of 2 or 3
First day handout (pdf) (course contract & exam schedule)

Tools
Online practice: calculus on the web (click `Linear Algebra` book icon)
Student companion site for the textbook
Linear Algebra Toolkit for practice, checking homework, etc.

### Homework Due Dates

 Due Date Required Reading & Suggested problems Turn-In Assignment* T 12/7 Final Exam. 10:30am-12:30pm, Perlstein 109. Chapter 7 practice for Final Exam. Material similar to this will appear on the final exam. Sect. 7.1: 1ab, 2ab, 3ab, 4ab, 5ab, 6ab, 7a, 8a, 9a, 10, 11, 12 Sect. 7.2: 1, 2, 8, 10, 11, 13, 16, 19 R 12/2 Read: Sect. 7.1-7.2 Recommended practice in Calculus on the Web (click the buttons according to the path below, and work problems online-warning: this list is not exhaustive-study your notes/text/hws too). (a) 21.Linear Algebra->3.Eigenvalues->1.Eigenvalues and Eigenvectors->Modules 1-3: Finding eigenvalues, Bases for eigenspaces, Eigenvectors in the plane Optional HW 13. (HW 13 will be graded for feedback, but the grade will not count. Turn in any or all problems. Pick it up in office hours 11:30am-1:30pm M 12/6. Sect. 6.1: 15a, 17, 19 Sect. 6.2: 6a, 13b, 15ab, 18c Sect. 6.3: 8, 9, 17a, 18, 20, 24abe Sect. 6.5: 6, 8 Sect. 6.6: 1 T 11/30 Read: Sect. 6.3-6.5, and pp.347-349 Recommended practice in Calculus on the Web (click the buttons according to the path below, and work problems online-warning: this list is not exhaustive-study your notes/text/hws too). (a) Inner Product Spaces. 21.Linear Algebra->4.Inner Products->Sections 1-3: Inner Product and Length, Orthogonal Vectors, Orthogonal Spaces T 11/23 Prepare: Exam 2 Recommended practice in Calculus on the Web (click the buttons according to the path below, and work problems online-warning: this list is not exhaustive-study your notes/text/hws too). (a) Linear combinations. 21.Linear Algebra > 2.Spaces and Transformations > 1.Linear Combinations (b) Linear independence. 21.Linear Algebra > 2.Spaces and Transformations > 2.Linear Independence (c) Matrix transformations. 21.Linear Algebra > 2.Spaces and Transformations > 3.Matrix Transformations (d) Row, column, null spaces. 21.Linear Algebra > 2.Spaces and Transformations > 4.Subspaces Exam 2: Chapter 4, 5.1-5.5 R 11/18 Read: Through Sect. 6.4 T 11/16 Read: Through Sect. 6.2 Quiz 2: Sect. 5.1-5.5 HW 12. Sect. 5.5: 10c Sect. 5.6: 8d-g, 12, 15 Sect. 6.1: 2cd, 4, 6, 8b R 11/11 Read: Through Sect. 6.1 T 11/9 Read: Through Sect. 5.6 HW 11. Sect. 5.4: 14, 23 Sect. 5.5: 4, 6c, 12a, 15 R 11/4 Read: Through Sect. 5.5 T 11/2 Warning: You must read and understand the examples in Section 5.3, as the lecture focus was very different. HW 10. Sect 5.3: 4, 8, 12, 15 Sect. 5.4: 6, 10, 22 R 10/28 HW 9. Sect. 4.4: 8, 13a Sect. 5.1: 2, 12, 20 (justify) Sect. 5.2: 2a-c, 7 (do 1 row reduction for a-d), 13, 17 T 10/26 Read: Through p.256 T 10/19 Read: Through p.243 HW 8. Sect. 4.3: 8ab, 18b R 10/14 Prepare: Exam 1 Exam 1 (Through Sect. 4.1) T 10/12 Read: Through Sect. 5.2 HW 7. Sect. 4.2: 8, 11, 14, 18a, 26, 28 (28 is mostly about verifying definitions and a characterization that is given to you) Sect. 4.3: 14a,16c R 10/7 Read: Through Sect. 4.4 Quiz 1. Chapters 1&2 11:25a-11:40a T 10/5 Read: Through Sect. 4.3 These problems are deferred until next time: Sect. 4.2: 4d, 6d, 8, 11, 14 HW 6. Sect. 3.4: 4a, 10a, 20, 30 Sect. 3.5: 8, 16, 40a Sect. 4.1: 16, 26 R 9/30 Read: Through p.186 of Sect. 4.2 T 9/28 Read: Through Sect. 3.5 HW 5. Sect. 2.3: 3, 9, 13, 15b (remember to show work), 16 Sect. 2.4: 17a Sect. 3.1: 8, 20 Sect. 3.2: 7, 9a, 12 Sect. 3.3: 4a, 5a, 17a, 21 R 9/23 Read: Sect. 2.4 through Sect. 3.2 (spend only a little time on 2.4) T 9/21 Read: Through Sect. 2.3 HW 4. Sect. 1.7: 9, 13, 14a, 22a, 24a Sect. 2.1: 12, 22, 23, 30, 31ab Sect. 2.2: 5, 13, 16 (just 5, not 4-7; but make it different than your previous work for 5) R 9/16 Read: Through Sect. 2.2 T 9/14 Read: Through Sect. 2.1 HW 3. Sect. 1.5: 5ab, 7a, 8d, 12, 17 Sect. 1.6: 2, 12ab, 20a, 21, 25 R 9/9 Read: Through page 90 T 9/7 Read: Through page 73 HW 2. Sect. 1.3: 6d, 7bc, 11, 12b, 18a, Sect. 1.4: 6a, 8, 10b, 12, 21a, 23, 29ab R 9/2 Read: Through page 59 T 8/31 Read: Through page 42 HW 1. Sect. 1.1: 8 Sect. 1.2: 5ab, 8ab, 13c, 17 R 8/26 Read: Through Section 1.3.

*Warning: Students with non-sanctioned editions of the textbook are responsible for working the problems as listed in the edition officially used in this course. The 9th edition with applications seems to be the same, but beware. The 10th edition has different problems, and who knows about the international editions.

 Date Class Activities and extra resources T 11/23 Exam 2, Sect. 4.1-5.5 R 11/18 6.2: Cauchy-Swarz inequality for inner product spaces; distance, dot product angle, and Pythagorean Theorem in inner product spaces; Orthogonal complement inner product subspaces come in pairs; nullspace and row space of a matrix are orthogonal complements with respect to the Euclidean inner product; 6.3: orthogonal bases and orthonormal bases; Expressing norm, distance, and inner product for an inner product space in terms of the Euclidean inner product by expressing vectors as coordinate vectors with respect to orthonormal bases; unique projection of a vector on an inner product subspace (Thms.6.3.4-5) T 11/16 6.1: Norm and distance defined from an inner product; the matrix inner product; an integral inner product; approximation of a continuous function f(x) from C[-pi,pi] by inner product projections on the functions 1, sin(x), and cos(x); Thm. 6.1.1.: properties of inner products (read, and think about dot products) R 11/11 5.6: (Various theorems about row/column/null spaces of A) Thm. 5.6.5: characterization of when Ax=b is consistent for a fixed b Thm. 5.6.6: characterization of when Ax=b is consistent for all b Thm. 5.6.7: #free parameters of solution to Ax=b when the system is consistent is #columns of A minus rank(A) Thm. 5.6.8: characterization of when Ax=0 has only the trivial solution 6.1: Definition of inner product from four axioms; inner products from weighted dot products; average of test scores as an inner product T 11/9 5.5: Row-equivalent matrices have linearly independent sets of vectors in the same columns, and bases in the same columns; a row-echelon form matrix R has rows with leading 1s as a basis for the row space, and columns with leading 1s as a basis for the column space; 5.6: Matrix rank and nullity; Four key vector spaces: row space, column space, nullspace, and nullspace of transpose; rank+nullity=#columns R 11/4 5.4: Every basis for a finite-dimensional vector space has the same size; Plus/Minus theorem to increase the size of a linearly independent set and keep independence, or remove a vector from a linearly dependent set and keep the same span; a spanning set of vectors can be reduced (if necessary) to a basis; a linearly independent set of vectors that does not span can be increased to a basis; a vector subspace has dimension at most that of the parent finite-dimensional subspace, with equal dimension iff the subspace equals the parent space; 5.5: Definitions of row space, column space, and nullspace of a matrix; Ax=b is consistent iff b is in the column space of A; Ax=b is consistent iff the solution x can be expressed as a particular solution plus a linear combination of the basis vectors of the nullspace of A; elementary row operations change neither the nullspace nor the row space of a matrix A T 11/2 5.4: Transforming between coordinates of the standard basis and an alternate basis via matrix multiplication; finite-dimensional vector spaces and dimension; examples of bases for finite- and infinite-dimensional vector spaces; all bases for a finite-dimensional vector space have the same number of vectors; computing a basis for the homogeneous solution set of a system of linear equations R 10/28 5.3: Geometric interpretation of linear dependence in R2, R3; more than n vectors in Rn are linearly dependent; 5.4: Plotting points in R2 via the standard basis and an alternate (skewed) basis; definition of basis (spans space, linearly independent); matrix invertibility criterion for a basis in Rn; interpretation of the linear transformation constructed using a basis of Rn; existence and uniqueness of the expression of every vector in terms of a linear combination of the basis; converting coordinate vectors between bases in Rn via matrix multiplication T 10/26 5.2: Vectors that span Pn; vectors that may or may not span R3, R4; a condition for two sets of vectors spanning the same space; 5.3: definition of linear dependence/independence of a set of vectors; use of Gaussian elimination to detect linear dependence/independence and find a largest subset of linearly independent vectors that span the same space R 10/21 5.2: Subspaces of vector spaces; subspace test; examples of subspaces; set of homogeneous solutions of a linear system of equations is a subspace; linear combinations of vectors; the span of a set of vectors is a subspace T 10/19 5.1: Axiomatic definition of real vector spaces; important examples of vector spaces; some properties deduced from vector space axioms R 10/14 Exam 1, Sect. 1.1-4.1 T 10/12 4.3: Linearity properties and use in characterizing linear transforms; formula for the standard matrix of a linear transform; projection onto a line is a linear transform; eigenvalues and eigenvectors of linear transforms derive from the associated standard matrix R 10/7 Quiz 1, Chapt. 1-2; 4.2: composition of linear transformations; 4.3: 1-1 linear operators on Rn are invertible T 10/5 4.1: Cauchy-Schwarz inequality used to prove n-dim triangle inequality; n-dim version of Pythagorean Theorem; dot product written as matrix multiplication; 4.2: coordinate functions Rn->R; linear transformations; standard matrix of a linear transformation; two methods to compute the standard matrix; projections, reflections, and rotations as linear operators; inverse pairs of linear transformations R 9/30 3.5: Intersection of 3 planes via Gaussian Elimination (Wolfram Demonstration); distance between a point and a plane, or between two planes; point-normal form, general form, vector form of a plane; general form, vector form of a line; parametric equations for a line; 4.1: Euclidean n-space, generalized from 2- and 3-space; Cauchy-Schwarz inequality T 9/28 3.3: Vector projections; distance between a point and a line; 3.4: Cross product in determinant and component form; area of a parallelogram from a cross product; scalar triple product and volume of a parallelepiped Vector projection, Cross product, Distance between a point and a line, Wolfram Demonstration (requires Mathematica or Mathematica Player) R 9/23 2.4: Combinatorial definition of the determinant, based on permutations and inversions; 3.1: Introduction to vectors; 3.2: Vector norms; 3.3: Vector dot product in geometric and component form; normal vector to a line Dot product, Wolfram Demonstration (requires Mathematica or Mathematica Player) T 9/21 2.3: Properties of the determinant, including det(AB)=det(A)det(B); A invertible iff det(A) is not zero; intro to eigenvalues and eigenvectors R 9/16 2.1: Determinants by cofactor expansion; matrix inverse via matrix adjoint; Cramer`s Rule; determinants of triangular matrices; 2.2: Determinants of elementary matrices; determinants by row reduction T 9/14 1.7: Diagonal, triangular, and symmetric matrices and their properties; 2.1: 2x2 and 3x3 determinants R 9/9 1.6: Number of solutions for Ax=b, including when A is invertible; easier detection of invertible matrices; extension of equivalent properties of an invertible matrix in Thm. 1.6.4 T 9/7 1.4: Matrix polynomials, properties of the matrix transpose; 1.5: Elementary matrices, equivalent properties of an invertible matrix, a method for computing A-1 using Gauss-Jordan elimination R 9/2 Matrix trace; properties of matrix arithmetic, but not commutativity of multiplication or multiplicative cancellation; zero matrix and identity matrix; definition and uniqueness of matrix inverse when it exists; 2x2 matrix inverses; matrix powers; proofs, including by induction and sequences of equations T 8/31 Homogeneous systems have infinitely many solutions when there are more variables than equations, and in other cases may or may not; various matrix definitions and operations, including addition, subtraction, dot product, multiplication; encoding a system as Ax=b; matrix multiplication as linear combinations of columns, or of rows; transpose and trace R 8/26 example with free and leading variables; variable order affects solution parameterization; properties of row-echelon and reduced row-echelon form; Gaussian elimination and Gauss-Jordan elimination algorithms; example algorithm instance using applet below; homogeneous systems are consistent and have at least the trivial solution T 8/24 linear equations and systems of linear equations; consistent and inconsistent systems; three cases for solution set; augmented matrix encoding; elementary row operations; introduction to row-echelon form, reduced row-echelon form, Gaussian elimination, and Gauss-Jordan elimination Practice Gaussian elimination or Gauss-Jordan elimination using this Java applet by Shing Hing Man.

page maintained by Robert Ellis / http://math.iit.edu/~rellis/