Math 332-001: Matrices
Instructor: Hemanshu Kaul
Office: 234B, Engineering 1
Phone: (312) 567-3128
E-mail: kaul [at]
Time: 9am, Monday-Wednesday-Friday.
Place: 026, Engineering 1.
Office Hours: 2pm-3pm, Monday, and 11am-12pm, Thursday. Also by
Tutoring Service: Academic
Course Topics and Textbook:
Topics: "Matrix algebra, rank, inverses; systems of linear
equations, determinants; vector spaces; inner product spaces;
eigenvalues and eigenvectors; linear transformations."
A detailed description of the lecture topics and the course
objectives is in the official course syllabus.
Textbook: Howard Anton, Elementary Linear Algebra, 9th ed.,
John Wiley and Sons
The Course Information Handout has the complete description of the student evaluation policy as well as other relevant information. Read it carefully! Updated!
- Friday, 12/8 : As requested, here are the three Exams.
- Thursday, 12/7 : All the select solutions are online.
- Friday, 12/1 : Homework assigned in the final week of the semester (12/4-12/8) need not be submitted. But, be sure to solve all the exercises and ask me for help if needed.
- Thursday, 11/16 : All the select solutions are online.
- Monday, 11/13 : Check the examination date and syllabus for Exam #3 below. Also, more select solutions will be uploaded below over the week. Be sure to ask me questions before Friday, 11/17, as I will be out of town that weekend.
- Friday, 10/27 : In addition to the usual office hours, I will be available from 1pm to 4pm in my office on both Monday and Tuesday for any help you might need with preparation for Exam #2. I might be out of office between 2pm and 3pm on Tuesday for a short while.
- Monday, 10/23 : Check the examination date for Exam #2 below. Also, more select solutions are being uploaded below.
- Wednesday, 10/11 : I will not be available for office hours tomorrow at 11am-noon. Instead, stop by between 3pm-4pm.
- Friday, 9/15 : Check the examination dates below. Also, more select solutions have been uploaded below.
- Wednesday, 9/13 : Please note the updated office location and office hours.
- Wednesday, 8/30 : The homework assignment due on Friday, 9/1, may be submitted on Wednesday, 9/6, if you wish.
- General: Check this webpage for homework assignments on Monday, Wednesday and Friday evenings.
- Exam # 1 : September 29, Friday. Topics: Chapters 1, 2, 4 (everything done in class till 9/20 Wednesday).
- Exam # 2 : November 1, Wednesday. Topics: Sections 4.2 to 5.4 (everything done in class from Monday, 9/25 till Wednesday, 10/25).
- Exam # 3 : November 20, Monday. Topics: Sections 5.4 to 6.3 (excluding Gram-Schmidt process).
- Final Exam : December 13, Wednesday, 8am to 10am. Topics: Everything done during the semester.
- Friday, 8/25 : Section 1.1 : Exercise 4b, 5b, 8, 11. Section 1.2: Exercise 3, 5a, 5d. Due Wednesday, 8/30. Also, read Sections 1.1 and 1.2. Select Solutions 1.
- Monday, 8/28 : Section 1.2 : Exercise 4c, 5b, 6c, 6d, 7c, 7d, 12, 13a, 17, 20. Due Friday, 9/1. Also, read Section 1.3. Select Solutions 2.
- Wednesday, 8/30 : Section 1.3 : Exercise 1, 2, 3g, 3i, 4e, 5k, 7b, 7c, 8b, 12a, 15a, 18a, 21, 27. Due Wednesday, 9/6. Select Solutions 3.
- Friday, 9/1 : Section 1.4 : Exercise 4b, 5, 7d, 10b, 13, 14, 20, 21a, 29. Due Friday, 9/8. Select Solutions 4.
- Wednesday, 9/6 : Section 1.5 : Exercise 1, 3, 5, 6c, 7b, 7c, 8d, 10, 15, 17, 21(use elementary matrices). Due Monday, 9/11. Also, read Section 1.6. Select Solutions 5.
- Friday, 9/8 : Section 1.6 : Exercise 4, 9a, 17, 20a (also attempt 28, not to be submitted), 21, 22a, 24, 27a. Due Wednesday, 9/13. Also, read Section 1.7. Select Solutions 6.
- Monday, 9/11 : Section 1.7 : Exercise 1a, 1b, 2b, 4, 6, 7, 15, 18, 20a, 20b, 22a, 24a. Due Friday, 9/15. Select Solutions 7.
- Wednesday, 9/6 : Section 2.2 : Exercise 2b, 3, 5, 9, 12a, 12b, 20a. Section 2.1 : Exercise 5, 9, 12, 17, 29. Due Monday, 9/18. Select Solutions 8.
- Friday, 9/15 : Section 2.3 : Exercise 4a, 4b, 5, 6, 9, 11, 13, 15b (i.e. do 15 for only 14b), 16. Try 19 (not to be submitted). Due Wednesday, 9/20. Also, read Chapter 3. Select Solutions 9.
- Monday, 9/18 : Section 4.1 : Exercise 2, 6def, 7, 15, 17, 19, 27. Due Friday, 9/22. Select Solutions 10.
- Wednesday, 9/20 : Section 4.1 : Exercise 10 (show how you found the vectors), 14cd, 18b, 22, 25, 32, 35a (show and justify the steps). Due Monday, 9/25. Select Solutions 11.
- Friday, 9/22 : No Homework
- Monday, 9/25 : Section 4.2 : Exercise 1bc, 2b, 4c, 5d, 9c, 14b, 19b, 20c, 21, 27, 29b, 30b. Due Monday, 10/2. Select Solutions 12.
- Wednesday, 9/27 : No Homework.
- Friday, 9/29 : Examination. No Homework.
- Monday, 10/2 : Section 4.3 : Exercise 2d, 4, 5ab, 6a, 7, 9cd, 10b, 11b, 20a. Due Friday, 10/6. Select Solutions 13.
- Wednesday, 10/4 : Section 4.3 : Exercise 12de, 13b, 14bc, 15ab, 18b, 19d, 21, 22a. Due Monday, 10/9. Select Solutions 14.
- Friday, 10/6 : Section 5.1 : Exercise 2, 3, 6, 7, 8, 10, 12, 17c, 18, 20. Due Wednesday, 10/11. Select Solutions 15.
- Monday, 10/9 : Section 5.1 : Exercise 5, 11, 15, 26. Due Friday, 10/13. Select Solutions 16.
- Wednesday, 10/11 : Section 5.2 : Exercise 1cde, 2bce, 3bcd, 4ac, 5abc. Be sure to show either a proof or a counterexample for each of the two conditions of Theorem 5.2.1. Due Monday, 10/16. Select Solutions 17.
- Friday, 10/13 : Section 5.2 : Exercise 6cf, 9b, 10a, 11bd, 12, 13, 14bd, 17. Due Wednesday, 10/18. Select Solutions 18.
- Monday, 10/16 : Section 5.1 : Exercise 29. Section 5.3 : Exercise 1cd, 3ac, 4bd, 8, 9, 11, 13, 16. Due Wednesday, 10/25. Select Solutions 19.
- Wednesday, 10/18 : No homework for submission. Make sure you understand the homework problems discussed in class.
- Monday, 10/23 : Section 5.3 : Exercise 14(how about more than n+1 vectors from P_n), 20de, 21bc. Due Friday, 10/27. Select Solutions 20.
- Wednesday, 10/25 : Section 5.4 : Exercise 1, 3bc, 4bc, 5, 6, 7b, 9b, 10b, 14, 16, 19, 20, one of (25, 26), 27bc. Due Monday, 10/30. Also, read Section 5.4 in its entirety. Select Solutions 21.
- Friday, 10/27 : Section 5.4 : Exercise 18cd, 21, 23. Due Friday, 11/3. Select Solutions 21part2.
- Monday, 10/30 : Section 5.5 : Exercise 2bc, 3c, 4, 5d, 6b, 7b, 11b, 12a, 13. Due Wednesday, 11/8. Change in Due Date. Select Solutions 22.
- Wednesday, 11/1 : Examination. No Homework.
- Friday, 11/3 : Section 5.5 : Exercise 3e, 6e, 9bc. Due Wednesday, 11/8. Select Solutions 22part2.
- Monday, 11/6 : Section 5.6 : Exercise [Not to be submitted: 3c, 4cd, 5b, 7bd, 8bd], [To be submitted: 2c, 6, 9, 10, 12b, 13, 14]. Due Friday, 11/10. Select Solutions 23.
- Wednesday, 11/8 : Section 6.1 : Exercise 6a, 7a, 10, 20, 21, 30. Due Monday, 11/13. Select Solutions 24.
- Friday, 11/10 : Section 6.1 : Exercise [Not to be submitted: 14, 15b, 17], [To be submitted: 9bc, 16ace, 22, 23, 26], Section 6.2 : Exercise [To be submitted: 2, 8a, 10b, one of (20, 21), 29]. Due Wednesday, 11/15. Select Solutions 24part2. Select Solutions 25.
- Monday, 11/13 : Section 6.2 : Exercise 4, 13b, 18ac, 19, 23, 25c. Section 6.3 : Exercise 5b, 6a, 7bc, 9b, 11b. Due Friday, 11/17. Select Solutions 25part2. Select Solutions 26.
- Wednesday, 11/15 : Section 6.3 : Exercise 10a, 14b, 15, 21, 26. Not to be submitted. Select Solutions 26part2.
- Friday, 11/17 : Section 6.3 : Exercise 17a, 20, 31.Due Monday, 11/27. Select Solutions 26part3.
- Monday, 11/20 : No HW (Examination).
- Wednesday, 11/22 : No HW.
- Monday, 11/27 : Section 6.3 : Exercise 24bdf. Section 6.5 : Exercise 8, 10, [not to be submitted: 6]. Due Friday, 12/1. Select Solutions 26part4. Select Solutions 27.
- Wednesday, 11/29 : Section 6.6 : Exercise 1, 3bc, 4a, 13. Due Monday, 12/4. Select Solutions 28.
- Friday, 12/1 : Section 7.1 : Exercise 1ad, 2ad, 3ad, 12, 16, 17, 20, 22, 23a. Due Wednesday, 12/6. Select Solutions 29.
- Monday, 12/4 : Section 7.2 : Exercise 1, 2, 7, 9, 10, 16, 22, 24. Not to be submitted. Select Solutions 30.
- Wednesday, 12/6 : Section 7.3 : Exercise 1bc, 4, 6, 7, 10, 11. Not to be submitted. Select Solutions 31.
- Friday, 12/8 : No HW.
- Friday, 8/25 : System of linear equations, Augmented Matrix, Elementary Row Operations, Row-Echelon form, Reduced Row-Echelon form. (From Sections 1.1 and 1.2)
- Monday, 8/28 : Gaussian Elimination, Gauss-Jordan Elimination, Homogeneous system of linear equations - trivial and non-trivial solutions, Matrix notation and terminology, Addition of matrices. (From Sections 1.2 and 1.3)
- Wednesday, 8/30 : Linear combination of matrices; Scalar product of matrices, Product of matrices - relation to dot product, column-by-column and row-by-row expressions, columns (rows) of the product as linear combination of columns (rows); Transpose of a matrix; Trace of a square matrix; Basic properties of matrix algebra. (From Section 1.3)
- Friday, 9/1 : Non-properties of Matrix multiplication, Zero matrices and their properties, Identity matrices and their properties, Invertible and Singular matrices, Inverse of 2X2 matrices, Inverse of product of invertible matrices, Inverse of transpose of an invertible matrix. (From Section 1.4)
- Wednesday, 9/6 : Integer powers of a matrix, Laws of exponents for matrices, Matrix polynomial, Properties of transpose, Elementary matrices - relation with row operations, inverse row operations, Statements equivalent to invertibility of a matrix, Method for finding inverse of a matrix. (From Sections 1.4 and 1.5)
- Friday, 9/8 : Number of solutions of a system of linear equations, Solving linear systems with matrix inversion, Solving linear systems with a common coefficient matrix, More statements equivalent to invertibility of a matrix, Invertible product of matrices implies the matrices are also invertible, Determining consistency. (From Section 1.6)
- Monday, 9/11 : Characterization of Invertible Matrices with proofs, Basic properties of Diagonal and Triangular matrices, and Symmetric matrices, Introduction to Determinants. (From Sections 1.7 and 2.2)
- Wednesday, 9/6 : Properties of determinant under row operations, Determinants of elementary matrices, Cofactor of an entry, Cofactor expansion for finding determinant, Adjoint of a matrix and the formula for the inverse, Cramers rule. (From Sections 2.2 and 2.1)
- Friday, 9/15 : Basic properties of determinants - scalar multiple of a matrix, sum of two matrices, Theorem 2.3.1, Invertibility in terms of determinant, Determinant of product of matrices, Determinant of the inverse, Characteristic equation, Eigenvalues, Eigenvectors. (From Section 2.3)
- Monday, 9/18 : Proof of "det(AB)= det(A) det(B)", Euclidean n-space - vectors, sum, scalar multiple, and their properties, Euclidean inner product and its properties, Norm and distance, Cauchy-Schwarz Inequality. (From Sections 2.3 and 4.1)
- Wednesday, 9/20 : Properties of length and distance, Dot product in terms of norm, Pythagorean Theorem in n-space, Matrix formulations for dot product, Matrix multiplication in terms of dot product. (From Section 4.1)
- Friday, 9/22 : Class canceled (to be made up later).
- Monday, 9/25 : Functions from R^n to R^m, Equivalence between linear transformations and multiplication with matrices, Zero, identity, reflection, projection, rotation, dilation, and contraction operators and their corresponding matrices, Compositions of linear transforms. (From Section 4.2)
- Wednesday, 9/27 : Compositions of linear transforms, Injective and surjective linear transforms, Characterization of invertible matrices in terms of their corresponding linear transforms. (From Sections 4.2 and 4.3)
- Friday, 9/29 : Examination.
- Monday, 10/2 : Inverse of a linear transform, Characterization of linearity. (From Section 4.3). Discussion of the Examination#1 problems.
- Wednesday, 10/4 : Proof of Characterization of linearity, Standard basis vectors for Euclidean spaces, Using the standard basis to find the standard matrix for any linear operator, Eigenvalues and eigenvectors for linear operators. (From Section 4.3)
- Friday, 10/6 : Definition of vector space, and examples. (From Section 5.1)
- Monday, 10/9 : More examples and non-examples of Vector Spaces, introduction to subspaces. (From Sections 5.1 and 5.2)
- Wednesday, 10/11 : Subspaces - examples and non-examples, Vector space of solution vectors of a homogenous system. (From Section 5.2)
- Friday, 10/13 : Linear combinations, Span of vectors is a subspace, Spanning sets for some vector spaces. (From Section 5.2)
- Monday, 10/16 : Spanning sets are not unique, Some elementary properties of vectors, Linear independence and dependence. (From Sections 5.2, 5.1, 5.3)
- Wednesday, 10/18 : Discussion of some old homework problems (#15 in 5.1, #13 + #17 + #11d in 5.2), Characterization of linear dependence and independence in terms of linear combinations. (From Sections 5.2, 5.1, 5.3)
- Friday, 10/20 : Fall Break
- Monday, 10/23 : Some simple reasons for linear dependence, Linear dependence in R^2 and R^3, A sufficient condition for linear dependence in R^n, Sufficient condition for linear independence of functions, Basis of a Vector Space, Standard bases for R^n and P_n. (From Sections 5.3, 5.4)
- Wednesday, 10/25 : Uniqueness of basis representation, coordinate vector relative to a basis, finite-dimensional vector spaces, Properties of sets with more or less vectors than a basis, Dimension of a finite-dimensional vector space. (From Section 5.4)
- Friday, 10/27 : Basis of the solution space of a homogenous system, Dimension of a subspace, Converting a spanning set or a linearly independent set into a basis, Plus/Minus theorem, How to extend a set of vectors into a basis. (From Section 5.4)
- Monday, 10/30 : Row space, Column space, and Null space of a matrix, Relation between consistency of a non-homogenous system and the Column space, General solution of a non-homogenous system in terms of a particular solution and a general solution of the corresponding homogenous system. (From Section 5.5)
- Wednesday, 11/1 : Examination.
- Friday, 11/3 : Finding Basis for Row(A), Col(A) and Null(A), Using Col(A) to find the basis of a Euclidean subspace. (From Section 5.5). Discussion of Exam #2.
- Monday, 11/6 : rank(A), nullity(A), Row(A)=Col(A^T), rank(A)=rank(A^T), rank + nullity = #of columns, rank and nullity in terms of the solution of the corresponding homogenous system, maximum value of rank, Consistency theorem, extension of characterization of invertible matrices. (From Section 5.6). Discussion of Exam #2.
- Wednesday, 11/8 : Overdetermined and Underdetermined linear systems and their properties, extension of characterization of invertible matrices, Inner product on a vector space, Inner product spaces, Inner products on R^n, Properties of Inner products. (From Sections 5.6 and 6.1)
- Friday, 11/10 : Unit sphere in an i.p.s., Relation between different inner products on R^n, Inner products on Matrices, Polynomials and Continuous functions, Cauchy-Schwarz inequality, Angle between two vectors in an i.p.s., Orthogonal vectors, Properties of length and distance. (From Sections 6.1 and 6.2)
- Monday, 11/13 : Properties of length and distance, Generalized Pythagoras Theorem, Orthogonal complement of a subspace, Properties and examples of Orthogonal complements, Orthogonal and Orthonormal sets of vectors, Orthonormal Basis. (From Sections 6.2 and 6.3)
- Wednesday, 11/15 : Orthonormal Basis, Coordinate vector relative to an Orthonormal basis, Norm, distance, and inner product using an orthonormal basis, Linear independence of orthogonal sets, Projection theorem, Orthogonal projection formulas. (From Section 6.3)
- Friday, 11/17 : Gram-Schmidt process for Orthonormal basis of an inner product space. (From Section 6.3) Review for Exam #3.
- Monday, 11/20 : Examination.
- Wednesday, 11/22 : Discussion of Exam #3.
- Monday, 11/27 : QR-decomposition, Change of basis problem and transition matrix. (From Sections 6.3 and 6.5)
- Wednesday, 11/29 : Relation between the two transition matrices, Orthogonal matrices, Orthonormal bases from an orthogonal matrix, Properties of orthogonal matrices, Transition matrix from one orthonormal basis to another, Eigenvalues and eigenvectors of a matrix, Characteristic polynomial and characteristic equation of a matrix, Eigenspace of a matrix w.r.t an eigenvalue, Eigenvalues of triangular matrices and positive integral powers of a matrix, Invertibility through eigenvalues. (From Sections 6.3, 6.5, 6.6, 7.1)
- Friday, 12/1 : Finding bases for the eigenspaces of a matrix, Definition and motivation for diagonalizability of matrices. (From Sections 7.1 and 7.2)
- Monday, 12/4 : Characterization of diagonalizable matrices, eigenvectors corresponding to distinct eigenvalues are linearly independent, Procedure for diagonalizing a matrix, Algebraic and geometric multiplicity of an eigenvalue - their relation to each other and to diagonalizability of the matrix, Orthogonal diagonalization of matrices. (From Sections 7.2 and 7.3)
- Wednesday, 12/6 : Orthogonal diagonalization of matrices, Characterization of Orthogonally diagonalizable matrices, Eigenvalues and eigenvectors of symmetric matrices, Procedure for orthogonal diagonalization of a matrix. (From Section 7.3) Discussion of final exam and related announcements.
- Friday, 12/8 : Final Exam review.
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