**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:** 203, Siegel Hall.

**Office Hours:** 3:05pm-4:15pm Monday, and 1pm-2pm 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, 12/1*: HW #12 has been uploaded. It is due by Friday, Dec 5th, 1pm.*Friday, 11/21*: Office hours on Tuesday, 11/25 will be held at 3:20pm-4:20pm instead of the usual time (1pm-2pm).*Friday, 11/21*: Make the obvious correction to the 1st line of the solution of 6.1.#23 in the HW#10 solutions I emailed you today.*Monday, 11/17*: After a vote in class, Exam #3 has been re-scheduled for Monday, December 1st. The exam syllabus will be announced on Wednesday, 11/19.*Wednesday, 10/29*: HW#9 will be announced on Wednesday, 11/5. It will be based on topics covered on 10/27, 10/29, and 11/5, and consequently, will be longer than usual.*Wednesday, 10/22*: Exam #2 is on Monday, November 3rd. The exam syllabus has been announced below.*Wednesday, 10/1*: Date for Exam #2 has been changed to Monday, Nov 3rd; see below.*Wednesday, 10/1*: HW#5 has been uploaded. See below.*Wednesday, 9/24*: HW#5 will be announced on Monday, 9/29. It will be based on topics covered on 9/22, 9/24, and 9/29.*Wednesday, 9/17*: EXAM #1 is in two weeks. It will be based on all topics covered till today. See below for the details.*Wednesday, 9/10*: HW #3 is a bit longer (but many problems have very short one-line solutions) than previous HWs. So, be sure to start working on it very soon.*Wednesday, 9/3*: Reminder: Weekly Homework gets uploaded every Wednesday evening.*Monday, 8/25*: Check this webpage regularly for homework assignments, announcements, etc.

*Exam # 1*: Wednesday, October 1st. Topics: All the topics covered in Chapters 1 and 2. This corresponds to the 7 lectures, the assigned reading HWs, and the assigned suggested/written HWs from 8/25 up to and including 9/17 (see HW and Class log for details). Graded exams with solutions & score-distribution distributed on 10/8.*Exam # 2*: Monday, November 3rd. Topics: All the topics corresponding to the HW#5, HW#6, HW#7, HW#8 covered in chapters 2, 4, and 5. This corresponds to the 9 lectures, the assigned reading HWs, and the assigned suggested/written HWs from 9/22 up to and including 10/22 (see HW and Class log for details). Graded exams with solutions & score-distribution distributed on 11/10.*Exam # 3*: Monday, December 1st. Topics: All the topics corresponding to the HW#9, HW#10, HW#11 covered in chapters 5 and 6. This corresponds to the 7 lectures, the assigned reading HWs, and the assigned suggested/written HWs from 10/27 up to and including 11/19 (see HW and Class log for details). Graded exams with solutions & score-distribution distributed on 12/3.*Final Exam*: Thursday, December 11th, 10:30am to 12:30pm. Topics: All topics studied during the semester, except Section 7.3 and Chapter 8.

You only have to submit solutions to

*Monday, 8/25*: Read Examples for Row-Echelon and Reduced Row-echelon forms from Section 1.2.Due Wednesday, 9/3. HW#1 solutions distributed in class on 9/3.__Homework #1__:

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

Written Problems: Section 1.1: #8. Section 1.2: #5b, #6c, #12, #13a, #17, #20.

*Wednesday, 8/27*: Read Section 1.3 up to Example 6.Due Wednesday, 9/10. HW#2 solutions distributed in class on 9/10.__Homework #2__:

Suggested Problems: Section 1.3: #1, #3, #4, #12b, #15a, #28.

Written Problems: Section 1.3: #1egh, #2, #7bc, #12a, #18a, #27.

*Monday, 9/8*: How can a row of AB be expressed as a linear combination of rows of B? (compare with what we did in class)

Read and prove the properties of transpose from Section 1.4 on pages 47-48. In particular, what is transpose(AB)? Can you prove it?Due Wednesday, 9/17. HW#3 solutions distributed in class on 9/17.__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: #7d, #10b, #13, #14, #21, #29. Section 1.5: #3bc, #5b, #7c, #8d, #10, #15, #17.

*Wednesday, 9/10*: Read and understand Theorem 1.4.10 on page 48 and its proof.

Think about problem #21 from Section 1.5.*Monday, 9/15*: Read and understand the proof of Theorem 1.6.3b as described in the class email.Due Wednesday, 9/24 (extended to Thursday, 9/25, 2pm). HW#4 solutions distributed through email on 9/25.__Homework #4__:

Suggested Problems: Section 1.6: #4, #5, #16, #21, #22a, #27. Section 1.7: #1ab, #2b, #4, #20, #24a. Section 2.1: #6, #8, #12. Section 2.2: #3, #9. Section 2.3: #11, #16.

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. Section 2.3: #5de, #8, #9, #13.

Optional Problem: Let A, B be square matrices of same size. Prove that: If AB is invertible then both A and B are invertible. (Hint: use properties of determinant.)*For your HW submission you can replace any 3 problems (at most ONE from each section) assigned above with this problem.*

*Wednesday, 9/17*: Read Example 3 on page 64; pay close attention to the final expression for**b**.

Read the statement of Theorem 1.7.1.

Do examples 3 and 4 on page 87 to practice co-factor expansion for calculating determinant.*Monday, 9/22*: 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, 10/8. HW#5 solutions distributed in class on 10/15.__Homework #5__:

Suggested Problems: Section 2.3: #15, Supplementary problems #5a and #7. Section 4.1:#2, #3, #6def, #14cd, #17b, #18b, #19, #27, . Section 4.2: #1, #2b, #5bd, #8-19, #20a, #29b.

Written Problems: Section 2.3: #15(only for 14(b)). Section 4.1: #7, #10a, #15b, #22, #25, #32(only part(b)), #35a. Section 4.2: #4c, #6d, #18b, #19b, #20c, #30a, #32.

__Extra-Credit Homework__:*Send me an email if you plan to do this HW.*Due Monday, 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 if any 0-1 matrix A of size n*n which has at least f(n)+1 such entries then it cannot be invertible.)*Monday, 10/6*: Read Examples 4&5 on pg200; Example 6 on pg202.Due Wednesday, 10/15. HW#6 solutions distributed through email on 10/16.__Homework #6__:

Suggested Problems: Section 4.3: #1, #3, #5ac, #7, #9c, #13 #15, #18, #19, #21,**#26**.

Written Problems: Section 4.3: #2d, #4, #6a, #9d, #10b, #11b, #12e, #14c, #18b, #20a, #22a, #27(give reasons).

*Monday, 10/6*: Read Examples 7&8 on pg204. Be sure to think about exercise#26 in Section 4.3.Due Wednesday, 10/22. HW#7 solutions distributed through email on 10/23.__Homework #7__:

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

Written Problems: Section 5.1: #5, #7, #8, #12, #15, #17ac, #18, #20, #28. Section 5.2: #1cd, #2ce, #3bc, #4ce, #5bc, #8ac, #9ac (do you see any similarity to #8ac?), #10a, #17.

*Wednesday, 10/22*: Be sure to write a proof of Theorems 5.3.1a and 5.3.2 on your own based on our discussion in class.

Read Example 4 on page 255 and Example 10 on page 259 in Section 5.4.

(Optional: Read and understand Theorem 5.3.4 and the two examples below it.)Due Wednesday, 10/29. HW#8 solutions distributed in class on 10/29. (send an email if you need a copy)__Homework #8__:

Suggested Problems: Section 5.2: #11-14. Section 5.3: #1, #3cd, #4ac, #8, #16. Section 5.4: #1, #6, #7b, #12, #14, #16, #18,

Written Problems: Section 5.2: #11c, #13, #14a. Section 5.3: #3ab, #4d, #9, #11, #13, #14. Section 5.4: #3b, #4a, #5, #9a, #10b, #15, #19b, #20, one of {#25, #26}, #27a.

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

*Wednesday, 10/29*: Read examples 7 and 8 on page 273 and compare them.Due Wednesday, 11/12. HW#9 solutions distributed through email on 11/13.__Homework #9__:

Suggested Problems: Section 5.4: #21, #36. Section 5.5: #2, #3, #5, #7b, #8. Section 5.6: #2, #5, #8, #13. Supplementary Exercises(pg. 290): #5, #10a, #12, #15.

Written Problems: Section 5.4: #23, #33. Section 5.5: #4, #5a, #6c, #9a, #11a, #12a (you don't need express non-basis vectors in terms of the basis), #13, #19. Section 5.6: #6, #10, #12b, #14. Supplementary Exercises(pg. 290): #6.

*Wednesday, 11/12*: Read Example 5 on page 299; Example 6 (and understand & verify statement (5)) on page 301; and Example 6 on page 313.Due Wednesday, 11/19. HW#10 solutions distributed through email on 11/21.__Homework #10__:

Suggested Problems: Section 6.1: #3, #4, #6a, #10, #14, #24, #26, #31. Section 6.2: #3, #8a, #13, #18b, #22, #24, #25c.

Written Problems: Section 6.1: #7a, #9bc, #15a, #16ace, one of {#20, #21}, #22, #23. Section 6.2: #2, #4, #10b, #18c, #19, #21, #23, one of {#28, #29}.

*Wednesday, 11/19*: Read Example 7 on page 324; and examples 1 and 2 on pages 335-336.Due Wednesday, 11/26. HW#11 solutions distributed in class and by email on 11/26.__Homework #11__:

Suggested Problems: Section 6.3: #2, #3, #8, #9, #12, #15, #16a, #19. Section 6.4: #1.

Written Problems: Section 6.3: #5b, #6a, #7b, #10b, #11a, #14a, #17a, #20, #21, #26, #29, #38. Section 6.4: #3a, #4a, #6, #14 (use the first sentence of Theorem 6.4.4). Supplementary Exercises: #8.

Due Friday, 12/5 before 1pm. [Submission is optional. Score in HW#12 may be used to replace one of your earlier HW scores.] HW#12 solutions distributed through email on 12/5.__Homework #12__:

Suggested Problems: 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 6.5: #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 (think about part b). Section 7.2: #2, #10, #12, #18, #24. Section 7.3: #6, #11.

*Monday, 8/25*: Discussion and 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, definitions of Row-Echelon and Reduced Row-echelon forms. (From Sections 1.1 and 1.2)*Wednesday, 8/27*: Discussion of course organization and purpose; examples of Row-Echelon and Reduced Row-echelon forms, leading and free variables, parametric form of infinite family of solutions, homogenous linear system and trivial solution, when does a homogenous system have a non-trivial solution and a simple outline of its proof, Gaussian and Gauss-Jordan elimination. (From Section 1.2 and elsewhere)*Monday, 9/1*: Labor Day holiday.*Wednesday, 9/3*: Matrix notation and terminology, Equality of two matrices, Addition and subtraction 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, Non-properties of Matrix multiplication, Basic properties of matrix algebra. (From Sections 1.3 and 1.4)*Monday, 9/8*: Linear combination of matrices, columns of the product as linear combination of columns, 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, Integer powers of a matrix, Laws of exponents for matrices, Properties of transpose. (From Section 1.4)*Wednesday, 9/10*: Transpose of AB with proof, Matrix polynomial, Elementary matrices - relation with row operations, inverse row operations, inverse of Elementary Matrix, Method for finding inverse of a matrix and its underlying logic, Number of solutions of a system of linear equations with proof, Solving linear systems with matrix inversion. (From Sections 1.4, 1.5, 1.6)*Monday, 9/15*: Discussion of solving Matrix equations/identities with regard to HW#2, Inverse of transpose of an invertible matrix, Five statements equivalent to invertibility of a matrix with proofs, Simpler condition for invertibility of a square matrix with proof. (From Sections 1.4, 1.5 and 1.6)*Wednesday, 9/17*: For sq matrices AB invertible implies A and B are invertible, Basic properties of Diagonal and Triangular matrices, and Symmetric matrices, 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, Invertibility in terms of determinant, Determinant of product of matrices, Determinant of the inverse. (From Sections 1.6, 1.7, 2.1, 2.2, 2.3)*Monday, 9/22*: Example for cofactor expansion for finding determinant, some more properties of determinants - sum of two matrices, Theorem 2.3.1, a short elementary proof of Cramer's rule, Characteristic equation, Eigenvalues and Eigenvectors.(From Sections 2.1, 2.3, and elsewhere.)*Wednesday, 9/24*: A handout with a short elementary proof of det(AB)=det(A)det(B) and related concepts and exercises was distributed. Euclidean n-space - vectors, sum, scalar multiple, and their properties, Euclidean inner product and its properties, Norm and distance, 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. (From Section 4.1 and elsewhere.)*Monday, 9/29*: 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 standard matrices, Composition of linear transforms and its relation to matrix multiplication. (From Section 4.2)*Wednesday, 10/1*: Exam #1.*Monday, 10/6*: Compositions of linear transforms, Injective and surjective 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. (From Section 4.3)*Wednesday, 10/8*: 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) Discussion of the Examination#1 problems.*Monday, 10/13*: Definition of vector space, examples and non-examples of Vector Spaces. (From Section 5.1)*Wednesday, 10/15*: 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, 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, 10/20*: Span(S) is the smallest subspace containing S, Spanning sets for some vector spaces, Spanning sets are not unique, Linear independence and dependence of vectors with examples and non-examples, relation between a vector equation and a linear system. (From Sections 5.2 and 5.3)*Wednesday, 10/22*: Characterization of linear dependence and independence in terms of linear combinations, Some simple reasons for linear dependence, Linear dependence in**R**^2 and**R**^3, 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, Uniqueness of basis representation, Coordinate vector relative to a basis, Dimension of a (finite-dimensional) vector space. (From Sections 5.3 and 5.4)*Monday, 10/27*: finite-dimensional vector spaces, Properties of sets with more or less vectors than a basis, Dimension of a finite-dimensional vector space, 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, Row space, Column space, and Null space of a matrix, Relation between consistency of a non-homogenous system and the Column space. (From Sections 5.4 and 5.5)*Wednesday, 10/29*: 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.(From Section 5.5)*Monday, 11/3*: Exam #2.*Wednesday, 11/5*: 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, maximum value of rank, 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. (From Section 5.6)*Monday, 11/10*: Distribution and discussion of Exam #2. Inner product on a vector space, Inner product spaces, Inner products on**R**^n, Relation between different inner products on**R**^n, Inner products on Matrices, Polynomials, and Continuous functions. (From Section 6.1)*Wednesday, 11/12*: 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)*Monday, 11/17*: Orthogonal and Orthonormal sets of vectors, Orthonormal Basis, Coordinate vector relative to an Orthonormal basis and the discussion of ideas underlying the proof, Normalization of an Orthogonal set, Norm, distance, and inner product using an orthonormal basis, Linear independence of orthogonal sets, Projection theorem, Orthogonal projection formulas. (From Section 6.3)*Wednesday, 11/19*: Gram-Schmidt process for creating an Orthonormal basis of an inner product space with proof, QR-decomposition, Best approximation in an R^3 and in any ips, Best approximation Theorem, Least squares problem, Derivation of a normal system from the least squares problem. (From Sections 6.3 and 6.4)*Monday, 11/24*: Problems that can solved using the least squares solution, 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, How to calculate positive-integral powers of a matrix. (From Sections 6.4, 6.5, and 6.6, and 7.1)*Wednesday, 11/26*: 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, 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, Procedure for diagonalizing a matrix, relation between P and D in the diagonalization. (From Sections 7.1 and 7.2)*Monday, 12/1*: Exam #3.*Wednesday, 12/3*: Distribution and discussion of Exam #3. How to check whether a matrix is not diagonalizable, Algebraic and geometric multiplicity of an eigenvalue - their relation to each other and to diagonalizability of the matrix, Orthogonal diagonalization of matrices, Characterization of Orthogonally diagonalizable matrices, Procedure for orthogonal diagonalization of a matrix. Overview of General Linear transformations between two vectors spaces. (From Sections 7.2 and 7.3; Chapter 8)