**Instructor:** Hemanshu Kaul

**Office:** 234B, Engineering 1

**Phone:** (312) 567-3128

**E-mail:** kaul [at]
math.iit.edu

**Time:** 1:50pm, Monday and Wednesday.

**Place:** 103, Engineering 1 Bldg.

**Office Hours:** 3:05pm-4:15pm Monday, and 2pm-3pm Tuesday, walk-ins, and by appointment. Emailed questions are also encouraged.

**Tutoring Service:** Mathematics tutoring at the Academic Resource Center.

**Online Problem Practice:** Linear Algebra book at COW (Calculus on Web).

|Course Information| |Advice| |Announcements| |Examinations| |Homework| |Class Log| |Links|

The

The official course syllabi: MATH 332.

Excellent advice by Doug West on how to write homework solutions for proof-based problems.

Why do we have to learn proofs?

Understanding Mathematics - a study guide

On a more abstract note, here is a discussion of Language and Grammar of Mathematics - which is what you are starting to learn in a course like this.

Excellent advice for math majors, especially those planning to go on to graduate school, by Terry Tao, 2006 Fields medallist. Required reading.

*Monday, 12/7*: Final Exam in two days. See details below.*Wednesday, 11/18*: No Office Hours next week (11/23 and 11/24).*Monday, 11/16*: Exam #3 in two weeks. See below for details.*Wednesday, 11/11*: Check your email for three important announcements sent today.*Monday, 10/26*: Exam #2 in one week. See below for details.

Homework #9 will be assigned on Wednesday, 11/4.*Monday, 10/4*: The due date for the Extra-credit HW has been changed to Tuesday, 10/13. See below.*Monday, 9/21*: Exam #1 in one week. See below.*Wednesday, 9/16*: For HW# 4 you may use the fact that "A is invertible if and only if det(A) is not 0" (Theorem 2.3.3) if needed.*Wednesday, 9/9*: Office Hours on Tuesday have been permanently changed to 2-3pm.*Wednesday, 9/2*: Due to Labor day weekend, Office Hours next week will held on Wednesday - 12:30-1:30pm and 3:05-4pm, instead of the usual Monday and Tuesday times.

And, if needed, you can submit HW#2 by 1pm on Thursday, 9/10.*Wednesday, 8/26*: Reminder: Weekly Homework gets uploaded every Wednesday evening.*Monday, 8/24*: Check this webpage regularly for homework assignments, announcements, etc.

*Exam # 1*: Monday, September 28th. Topics: All the topics corresponding to the HW#1, HW#2, HW#3, HW#4. This corresponds to the 7 lectures, the assigned reading HWs, and the assigned suggested/written HWs from 8/24 up to and including 9/16 (see HW and Class log for details).*Exam # 2*: Monday, November 2nd. Topics: All the topics corresponding to the HW#5, HW#6, HW#7, HW#8. This corresponds to the 8 lectures, the assigned reading HWs, and the assigned suggested/written HWs from 9/21 up to and including 10/21 (see HW and Class log for details).*Exam # 3*: Wednesday, December 2nd. Topics: All the topics corresponding to the HW#9, HW#10, HW#11. This corresponds to the 7 lectures, the assigned reading HWs, and the assigned suggested/written HWs from 10/26 up to and including 11/18 (see HW and Class log for details).*Final Exam*: Wednesday, December 9th, 2pm to 4pm. Topics: All topics studied during the semester.

You only have to submit solutions to

However, solving a majority of the

*Wednesday, 8/24*: Read Examples for Row-Echelon and Reduced Row-echelon forms from Section 1.2.

Read Section 1.3 up to Example 6.Due Wednesday, 9/2. HW#1 solutions distributed in class on 9/9.__Homework #1__:

Suggested Problems: Section 1.1: #4b, #5, #11. Section 1.2: #3, #5ad, #6d, #13c, #20. Section 1.3: #1.

Written Problems: Section 1.1: #8. Section 1.2: #5b, #6c, #12, #13a, #17. Section 1.3: #1egh, #2.

*Monday, 8/31*: Read examples in Section 1.3

Find A and B such that AB is not equal to BA.Due Wednesday, 9/9 (extendible to 1pm, Thursday, 9/10). HW#2 solutions distributed in class on 9/16.__Homework #2__:

Suggested Problems: Section 1.3: #3, #4, #12b, #15a, #25, #28. Section 1.4: #4b, #5, #7, #17, #20, #31. Section 1.5: #1, #5ac, some parts of (#6, #7, #8), #9.

Written Problems: Section 1.3: #7bc, #12a, #18a. Section 1.4: #7d, #13, #14, #21b, #29. Section 1.5: #3bc, #5b, #7c, #8d.

*Wednesday, 9/2*: Read Examples #4 and #5 in section 1.5 (pg. 55-57).Due Wednesday, 9/16. HW#3 solutions distributed in class on 9/21.__Homework #3__:

Suggested Problems: Section 1.4: #4b, #5, #7, #17, #20, #31. Section 1.5: #1, #5ac, some parts of (#6, #7, #8), #9, #21.

Written Problems: Section 1.4: #7c, #10b, #36. Section 1.5: #10, #15, #17.

*Monday, 9/14*: Read Examples #3 and #4 in Section 1.6 (pg. 64-65); pay close attention to the final expression for**b**

Read and understand the Theorem 1.7.1 on page 70.Due Wednesday, 9/23. HW#4 solutions distributed in class on 9/23.__Homework #4__:

Suggested Problems: Section 1.6: #4, #5, #16, #21, #22a, #27. Section 1.7: #1ab, #2b, #4, #7, #20, #24a. Section 2.1: #6, #8, #12. Section 2.2: #3, #9.

Written Problems: Section 1.6: #9(main question before the parts), #17, #20a, #24, #28a. Section 1.7: #6, #7, one of (#15b, #18). Section 2.1: #5, #7, #16, #22. Section 2.2: #5, #12ab, #20a.*Wednesday, 9/16*: Do Examples 3 and 4 on page 87 to practice co-factor expansion for calculating determinant.*Monday, 9/21*: Browse through Chapter 3 to review topics, such as geometric interpretation, dot and cross product, distance and norm, projection, etc., about vectors in**R**^{2}and**R**^{3}that you learnt in Calculus III. In class we will study the same topics (and more) for vectors in any**R**^{n}from Chapter 4.Due Wednesday, 10/7. HW#5 solutions distributed in class on 10/14.__Homework #5__:

Suggested Problems: Section 2.3: #5de, #6, #11, #15, #16, Supplementary problems #5a and #7. Section 4.1:#2, #3, #6def, #14cd, #17b, #18b, #19, #22, #26, #27. Section 4.2: #1, #2b, #5bd.

Written Problems: Section 2.3: #8, #9, #13, #15(only for 14(b)). Section 4.1: #7, #10a, #15b, #25, #32(only part(b)), #35a. Section 4.2: #4c, #6d.

*Monday, 9/30*: Read the descriptions of the reflection, rotation, projection, rotation, and dilation operators given in Tables 2, 3, 4, 5, 6, 8, 9 on pg 185-190.__Extra-Credit Homework__:*Send me an email if you plan to do this HW.*Due 2pm, Tuesday, 10/13. Solutions can be discussed in person.

Problem #1: Solve the exercises given at the end of the class handout with a short proof of "det(AB)=det(A)det(B)".

Problem #2: Let A be a square matrix of size n whose all n^2 entries are either 0 or 1 (called 0-1 matrix).

(a) If A is invertible, then what is the maximum number of entries of A that can be 0?

(b) If A is invertible, then what is the maximum number of entries of A that can be 1?

(Note: For each of the parts above, you will have to conjecture some answer in terms of n (a formula, f(n), that lies between 0 and n^2). Then construct a 0-1 matrix A of size n*n which is invertible and has f(n) such entries. AND prove that any 0-1 matrix A of size n*n which has at least f(n)+1 such entries cannot be invertible.)*Wednesday, 10/7*: Read Examples 4 and 5 on Pg 200.Due Wednesday, 10/14. HW#6 solutions distributed in class on 10/21.__Homework #6__:

Suggested Problems: Section 4.2: #1, #2b, #5bd, #8-#20, #29b. Section 4.3: #1, #3, #5ac, #7, #9c,**#26**.

Written Problems: Section 4.2: #18b, #19b, #32. Section 4.3: #2d, #4, #6a, #9d, #10b, #11b, #20a, #22a, #27(give reasons).

*Wednesday, 10/14*: Read Examples 7-8 on pages 204-205. Read Examples 1-7 on pages 223-226.Due Wednesday, 10/21. HW#7 solutions distributed in class on 10/26.__Homework #7__:

Suggested Problems: Section 4.3: #13 #15, #18, #19, #21. Section 5.1: #1-3, #6, #10, #11, #13, #26.

Written Problems: Section 4.3: #12e, #14c, #18b(just the direct calculation). Section 5.1: #5, #7, #8, #12, #15, #17ac, #18, #20.

*Wednesday, 10/21*: Read Example 7 on pages 233-234. Read Examples 11-12 on pages 236-237.Due Wednesday, 10/28 [Strict Deadline]. HW#8 solutions distributed on 10/28.__Homework #8__:

Suggested Problems: Section 5.1: #1-3, #6, #10, #11, #13, #26. Section 5.2: #1abe, #2ab, #3a, #4, #5d, #11-#13, #26.

Written Problems: Section 5.1: #28. Section 5.2: #1cd, #2ce, #3bc, #4ce, #5bc, #8ac, #9ac (do you see any similarity to #8ac?), #10a, #11c, #13, #14a, #17.

Due Wednesday, 11/11. HW#9 solutions distributed on 11/18.__Homework #9__:

Suggested Problems: Section 5.3: #1, #3cd, #4ac, #8, #16. Section 5.4: #1, #6, #7b, #12, #14, #16, #18, #21, #36. Supplementary Exercises(pg. 290): #5, #10a, #12, #15.

Written Problems: Section 5.3: #3ab, #4d, #9, #11, #13, #14. Section 5.4: #3b, #4a, #5, #15, #19b, #20, #23, #33. Supplementary Exercises(pg. 290): #6.

Due Wednesday, 11/18 [Strict Deadline]. HW#10 solutions distributed on 11/18.__Homework #10__:

Suggested Problems: Section 5.4: #21, #36. Section 5.5: #2, #3, #5.

Written Problems: Section 5.4: #9a, #10b, one of {#25, #26}, #27a. Section 5.5: #4, #5a, #6c.

Note that #25 and #26 in Section 5.4 illustrate how a question about vectors in a general vector space of dimension n can be converted into a question about the corresponding coordinate vectors in**R**^n.

Due Monday, 11/23 [Strict Deadline]. HW#11 solutions distributed on 11/23.__Homework #11__:

Suggested Problems: Section 5.5: #2, #3, #5, #7b, #8. Section 5.6: #2, #5, #8, #13. Section 6.1: #3, #4, #6a, #10, #14, #24, #26, #31.

Written Problems: Section 5.5: #9a, #11a, #12a (you don't need express non-basis vectors in terms of the basis), #13, #19(hint: are A and AB row equivalent?). Section 5.6: #6, #10, #12b (hint: first calculate for what values of t is rank(A)=3), #14. Section 6.1: #7a (read formula(5) in example 6), #9bc, #15a, #16ace, one of {#20, #21}, #22, optional(#23).

Due Monday, 12/7. [Submission is optional. Score in HW#12 may be used to replace one of your earlier HW scores.] HW#12 solutions distributed on 12/7.__Homework #12__:

Suggested Problems: Section 6.2: #3, #8a, #13, #18b, #22, #24, #25c. Section 6.3: #2, #3, #8, #9, #12, #15, #16a, #19. Section 6.5: #1-4, #7c, #9. Section 6.6: #1. Section 7.1: #7&8&9, #12, #22bc, #24, #25. Section 7.2: #1, #5, #9, #14.

Written Problems: Section 6.2: #2, #4, #10b, #18c, #19, one of {#21, #23}, one of {#28, #29}. Section 6.3: #5b, #6a, #7b, #10b, #11a, #14a, #17a, #20, #21, #26, #29, #38. Section 6.5: #7ab, #8ab, #10abc, #18. Section 6.6: #3bc, #4, #13. Section 7.1: #1&2&3 part(a), #11, #14a, One of {#20, #21}, #22a, #23a. Section 7.2: #2, #10, #12, #18, #24.

*Monday, 8/25*: Discussion of course organization and purpose; Course survey; linear equations and systems of linear equations, comparison to lines and planes, consistent and inconsistent systems, only three possibilities for number of solutions of a linear system, Augmented matrix, Elementary row operations and back substitution for solving linear systems. (From Sections 1.1 and 1.2)*Wednesday, 8/27*: Definitions of Row-Echelon and Reduced Row-echelon forms, examples of Row-Echelon and Reduced Row-echelon forms, leading and free variables, parametric form of infinite family of solutions, Gaussian and Gauss-Jordan elimination, homogenous linear system and trivial solution, when does a homogenous system have a non-trivial solution and a simple outline of its proof, Matrix notation and terminology, Equality of two matrices, Addition and subtraction of matrices. (From Sections 1.2, 1.3, and elsewhere)*Monday, 8/31*: Scalar product of matrices, Product of matrices - condition for definition, relation to dot product, column-by-column and row-by-row expressions, columns (rows) of the product as linear combination of columns (rows), Matrix equation and its relation to system of linear equations, Transpose of a matrix, Trace of a square matrix, Basic properties of matrix algebra, Non-commutativity of Matrix multiplication. (From Sections 1.3 and 1.4)*Wednesday, 9/2*: Non-properties of Matrix multiplication - Cancelation law and commutativity of product, Zero matrices and their properties, Identity matrices and their properties, Invertible and Singular matrices, Uniqueness of the inverse, Inverse of 2X2 matrices, Inverse of product of invertible matrices, Integer powers of a matrix, Laws of exponents for matrices, Elementary matrices - relation with row operations, Method for finding inverse of a matrix and its underlying logic. (From Sections 1.3, 1.4, and 1.5)*Monday, 9/7*: Labor Day Holiday.*Wednesday, 9/9*: Discussion of HW#1. Properties of transpose, Transpose of AB, Inverse of transpose of an invertible matrix, Inverse of Elementary Matrix and their relation to inverse row operations, Three statements equivalent to invertibility of a matrix with proofs. (From Sections 1.4, and 1.5)*Monday, 9/14*: Number of solutions of a system of linear equations with proof, Solving linear systems with matrix inversion, Simpler condition for invertibility of a square matrix with proof, for sq matrices AB invertible implies A and B are invertible, Two more statements equivalent to invertibility of a matrix with proofs, Basic properties of Diagonal and Triangular matrices, and Symmetric matrices. (From Sections 1.6, and 1.7)*Wednesday, 9/16*: Discussion of HW#2. Introduction to Determinants, Properties of determinant under row operations, determinants of triangular matrices and matrices with a zero row or column, det(A)=det(transpose(A)), determinants of elementary matrices, Cofactor of an entry, Cofactor expansion for finding determinant, Adjoint of a matrix and the formula for the inverse, Cramer's rule. (From Sections 2.1 and 2.2)*Monday, 9/21*: Discussion of HW#3. A short elementary proof of Cramer's rule, Invertibility in terms of determinant with proof, Determinant of product of matrices with proof, Determinant of the inverse with proof. (From Sections 2.1, 2.3, and elsewhere.)*Wednesday, 9/23*: Discussion of HW#4. A handout with a short elementary proof of det(AB)=det(A)det(B) and related concepts and exercises was distributed. Characteristic equation, Eigenvalues and Eigenvectors, 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, 4.1 and elsewhere.)*Monday, 9/28*: Mid-term Exam #1.*Wednesday, 9/24*: Cauchy-Schwarz Inequality, Properties of length and distance - including their triangle inequalities with proof, Dot product in terms of norm of sum and difference, Pythagorean Theorem in n-space, Matrix formulations for dot product, Functions from**R**^n to**R**^m, Equivalence between linear transformations and multiplication with matrices, Zero, identity, reflection operators and their corresponding standard matrices. (From Sections 4.1 and 4.2)*Monday, 10/5*: Detailed discussion of Exam#1 solutions and performance. Reflection, projection, rotation, dilation, and contraction operators and their corresponding standard matrices. (From Section 4.2)*Wednesday, 10/7*: Compositions of linear transforms, Injective and surjective(onto) linear transforms, Characterization of invertible matrices in terms of their corresponding linear transforms, Inverse of a linear transform - when does it exist and how to find it, Characterization of linearity with proof using standard Euclidean basis vectors to form the standard matrix. (From Section 4.3)*Monday, 10/12*: Fall Break.*Wednesday, 10/14*: Discussion of HW#5 problems. 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; Definition of vector space, examples and non-examples of Vector Spaces. (From Sections 4.3 and 5.1)*Monday, 10/19*: Examples (**R**^n, M_{m x n}, F[a,b],**P**_n) and non-examples (R^2 with non-standard scalar multiplication, Polynomials of degree=n, Non-negative quadrant of R^2, Invertible Matrices) of Vector Spaces, Some elementary properties of vector spaces with proofs. (From Section 5.1)*Wednesday, 10/21*: Some elementary properties of vector spaces with proofs, introduction to subspaces with examples and non-examples, Characterization of subspaces, Vector space of solution vectors of a homogenous system (Null(A)), Linear combination of vectors, When is vector in**R**^n a linear combination of some other vectors in**R**^n? - conversion to a linear system, Span of vectors, Span(S) is a subspace. (From Section 5.2)*Monday, 10/26*: Discussion of HW#6, HW#7, HW#8. Spanning sets for some vector spaces and subspaces, Conversion of a spanning set problem into a linear system problem. (From Sections 5.2 and 5.3)*Wednesday, 10/28*: Discussion of HW#7 and HW#8 - examples and non-examples of vector spaces and subspaces. Spanning sets for some subspaces, further comments on Conversion of a spanning set problem into a linear system problem, linear independence and its motivations. (From Sections 5.2 and 5.3)*Monday, 11/2*: Mid-term Exam #2.*Wednesday, 11/4*: Linear independence and dependence of vectors with examples and non-examples, relation between a vector equation and a linear system, Characterization of linear dependence and independence in terms of linear combinations, Some simple reasons for linear dependence, A sufficient condition for linear dependence in**R**^n, Basis of a Vector Space, Standard bases for**R**^n,**P**_n, and**M**_nn, how to show S is a Basis of**R**^n, Dimension of a (finite-dimensional) vector space, Properties of sets with more or less vectors than a basis, . (From Sections 5.3 and 5.4)*Monday, 11/9*: Standard bases for**R**^n,**P**_n, and**M**_nn, how to show S is a Basis of**R**^n,**P**_n, etc., Basis of the solution space of a homogenous system, Uniqueness of basis representation, Coordinate vector relative to a basis with examples from**R**^n and**P**_n, Properties of sets with more or less vectors than a basis, Dimension of a subspace, Converting a spanning set or a linearly independent set into a basis. (From Section 5.4)*Wednesday, 11/11*: Detailed discussion of Exam#1 solutions and performance. 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 for R^n, Row space, Column space, and Null space of a matrix. (From Sections 5.4 and 5.5)*Monday, 11/16*: 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, Row operations and Row, Col and Null spaces of a matrix and their bases, Finding Basis for Row(A), Col(A) and Null(A), Using Col(A) to find the basis of a Euclidean subspace, Statements with proofs related to: 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. (From Sections 5.5 and 5.6)*Wednesday, 11/18*: Consistency theorem, overdetermined and underdetermined linear systems and their properties, Consistency properties of linear systems with non-square coefficient matrices, extension of characterization of invertible square matrices, Inner product on a vector space, Inner product spaces, Inner products on**R**^n, Relation between different inner products on**R**^n [Read equation (1) and Example 2], Inner products on Matrices, Polynomials, and Continuous functions. (From Sections 5.6 and 6.1)*Monday, 11/23*: Properties of Inner products, Cauchy-Schwarz inequality with proof, Angle between two vectors in an i.p.s., Orthogonal vectors, Properties of length (norm) and distance, Generalized Pythagoras Theorem, Orthogonal complement of a subspace, Properties and examples of Orthogonal complements, Null(A) and Row(A) are orthogonal complements, Finding the basis of an orthogonal complement in the Euclidean space, Orthogonal and Orthonormal sets of vectors, Orthonormal Basis. (From Sections 6.1, 6.2 and 6.3)*Wednesday, 11/25*: Thanksgiving Break*Monday, 11/30*: Coordinate vector relative to an Orthonormal basis, Normalization of an Orthogonal set, Norm, distance, and inner product using an orthonormal basis, Linear independence of orthogonal sets, Projection theorem, Orthogonal projection formulas, Gram-Schmidt process for creating an Orthonormal basis of an inner product space with proof, Change of basis problem and transition matrix for relating the two coordinate vectors, Relation between the two transition matrices with proof, Orthogonal matrices, Orthonormal bases from an orthogonal matrix, Elementary properties of orthogonal matrices, Transition matrix from one orthonormal basis to another. (From Sections 6.3, 6.4, 6.5, and 6.6)*Wednesday, 12/2*: Mid-term Exam #3.*Monday, 12/7*: Distribution and discussion of Exam #3. Positive integral powers of a matrix, Review of :Eigenvalues and eigenvectors of a matrix, Characteristic polynomial and characteristic equation of a matrix; Eigenspace of a matrix w.r.t an eigenvalue, Finding bases for the eigenspaces of a matrix, Definition and motivation for diagonalizability of matrices, Characterization of diagonalizable matrices in terms of eigenvectors and sum of nullities, eigenvectors corresponding to distinct eigenvalues are linearly independent, How to check whether or not a matrix is diagonalizable, Procedure for diagonalizing a matrix, relation between P and D in the diagonalization. (From Sections 7.1 and 7.2)