Instructor: Hemanshu Kaul

Office: 125C, Rettaliata Engg.
Phone: (312) 567-3128
E-mail: kaul [at] iit.edu

Time: 11:25am, Tuesday and Thursday.
Place: 152, Pritzker Science Center.

Office Hours: 1:15pm-2:15pm Tuesday and Thursday, and by appointment (send email).
Emailed questions are also encouraged.
As well as the discussion forum at Piazza.

Math TA Office Hours: Quinn Stratton at 10am-12pm on Mondays and 4:45pm-5:45pm on Tuesdays in 129, Retalliata Engg.
ARC Tutoring Service: Mathematics tutoring at the Academic Resource Center.

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

• Thursday, 10/24 : Note the Final Exam date below.


• Tuesday, 9/10 : All the Exam dates have been announced below.
• Tuesday, 8/20 : Check this webpage regularly for homework assignments, announcements, etc.

• Exam #1 : Thursday, 9/26. Topics: All the topics corresponding to the HW#1, HW#2, HW#3, HW#4.
• Exam #2 : Thursday, 10/24. Topics: All the topics corresponding to the HW#5, HW#6, HW#7.
• Exam #3 : Tuesday, 11/19. Topics: All the topics corresponding to the HW#8, HW#9, HW#10.
• Final Exam : Tuesday, 12/3, 10:30am-12:30pm. Topics: All topics studied during the semester.

• Tuesday, 8/20 : Read Example 6 in Section 1.1 and examples for Row-Echelon and Reduced Row-echelon forms in Section 1.2.


• Thursday, 8/22 : Read examples 1 to 6 in Section 1.3
  Find 2x2 (and 3x3) matrices A and B such that AB is not equal to BA.
  Read about Partitioned Matrices and Examples 7, 8, 9, 10, 11 and 12 in Section 1.3.


• Homework #1 : Due Thursday, 8/29. HW#1 solutions distributed in class on 8/29.
  Suggested Problems: Section 1.1: 1, 5, 7, 9, 21&26, TF. Section 1.2: 1, 3, 15, 19, 35, TF. Section 1.3: 23, 30.
  Submission Problems: [Comment: When solving a system, set up the augmented matrix and then apply row operations; clearly label each row operation applied and show all intermediate steps.]. Section 1.1: 12, 16b, 20b, TF(e)(f)(g). Section 1.2: 18, 24ac, 26, 31, 34, 43a. Section 1.3: 27, 30a.



• Thursday, 8/29 : Read Definition 6, Theorem 1.3.1, and examples 8,9,10 in Section 1.3.
  Read Theorem 1.4.5 and Examples 7, 8 in Section 1.4.


• Homework #2 : Due Thursday, 9/5. HW#2 solutions distributed in class on 9/5
  Suggested Problems: Section 1.2: 13, 33, . Section 1.3: 1, 3, 5, 7, 8af, 11, 13, 15, 23, 25, 29, 32, TF(e)(f)(l)(m).
  Written Problems: Section 1.2: 8 and 12 (solve them in continuation/ together), 38, 39, 40, TF(b)(d)(g)(i). Section 1.3: 5de, 16, 36b, TF(m). Section 1.4: 54.
  



• Tuesday, 9/3 : Read Definition 7 and Example 11 in Section 1.3 followed by Theorems 1.4.8 and 1.4.9 in Section 1.4.


• Thursday, 9/5 : Read Examples #4 and #5 in Section 1.5; Examples #3 and #4 in Section 1.6 and pay close attention to the final expression for b
  Read Theorem 1.6.4 in Section 1.6.
  


• Homework #3 : Due Thursday, 9/12. HW#3 solutions distributed in class on 9/12
  Suggested Problems: Section 1.4: 3, 4, 5, 9, 10, 12, 13, 17, 23, 24, 32, 35, 41, 44, 45, 49. Section 1.5: #1, #3, #5, #7, #9, #13, #19, #25, #27, #29, #33, TF.
  Written Problems: Section 1.3: 8af. Section 1.4: 31(b)(c), 33(a), 36, 40, 43, 50, TF(j)(k). Section 1.5: #6b, #8c, #16, #20a, #22, #28.
  



• Tuesday, 9/10 : Read Examples #3 and #4 in Section 1.6 and pay close attention to the final expression for b
  Read Theorem 1.6.4 in Section 1.6 and Theorem 1.7.1 in Section 1.7.
  


• Thursday, 9/12 : Do Examples 3-5 from Section 2.1 to practice co-factor expansion for calculating determinant.
  Do example 5 in Section 2.2 for a combination of Row operations and Cofactor expansion.


• Homework #4 : Due Thursday, 9/19. HW#4 solutions distributed in class on 9/19.
  Suggested Problems: Section 1.6: #5, #15, #18a, #19, #21, T/F#(b)(d). Section 1.7: #1, #9, #13, #17, #21, #25, #30, #37, #47. Section 2.2: #5, #11, #17, #23, #27, #29, #33, TF(d).
  Written Problems: Section 1.5: #32. Section 1.6: #14, #20, #22, T/F#(f). Section 1.7: #26 (just set up the system without finding the exact values for a,b,c), #28, #34b. Section 2.2: #20, #24, #26, #34. Section 2.3: #34. Chapter 2 Supplementary Problems: #33.



• Tuesday, 9/17 : Do Example 1 and Example 3 in Section 9.1 for an example of direct construction of L and U from the Gaussian elimination procedure.
  Read the description of Network Analysis and do Examples 1 and 2 in Section 1.9 for an example of network flows.


• Thursday, 9/19 : Read Examples 6 and 8 in Section 4.1.


• Homework #5 : Due Thursday, 10/3. HW#5 solutions distributed in class on 10/3.
  Suggested Problems: Section 2.3: #7, #9, #17, #18, #33, #36, #37, TF(abcdghij). Section 9.1: #1, #5, TF(a)(b). Section 4.1: #1, #2, #5, #6, #9, #10, #12.
  Written Problems: Section 9.1: #6. Section 4.1: #4, #7, #8, #16, #19and20, 22, and these two problems.


• Tuesday, 10/1 : Read Example 6, and Examples 7-10 together with the Figure of function spaces in Section 4.2.
  Read Theorem 4.2.4 in Section 4.2.


• Thursday, 10/3 : Read Examples 1-5 in Section 4.3.


• Homework #6 : Due Thursday, 10/10. HW#6 solutions distributed on 10/10.
  Suggested Problems: Section 4.2: #1, #2, #3, #4, #5, #7, #9, #11, #13, #16, #17, T/F.
  Written Problems: Section 4.2: #1cd, #2abe, #3b, #4b, #9a, #10a, #12c, #18, TF(g)(h). Section 4.3: #4a, #6, #10a.
  


• Thursday, 10/10 : Read Definition 2 and Example 7 in Section 4.3.
  Read Definitions 1 and 2, and Examples 1-4 and 7-8 in Section 4.4.


• Homework #7 : Due Thursday, 10/17. HW#7 solutions distributed on 10/17.
  Suggested Problems: Section 4.3: #1, #2, #3, #5, #11, #26. Section 4.4: #3, #5, #7, #8, #9, #13, #14, #15, #17, #20, #TF.
  Written Problems: Section 4.3: #11, #22, #27, #28, TF(e). Section 4.4: #4, #6, #10 [HINT: cos^2(x) - sin^2(x) = cos(2x)], #16, #25.
  


• Tuesday, 10/22 : Read Example 3 in Section 4.6.
  Read Examples 2, 3, 4, 5, 9 in Section 4.7.


• Homework #8 : Due Thursday, 10/31. This is a longer HW based on 3 lectures. HW#8 solutions distributed on 10/31.
  Suggested Problems: Section 4.5: #3, #8, #9, #11, #TF. Section 4.6: #1, #3, #5, #9, #13, #14, #16. Section 4.7: #3, #5, #8a, #11, #14, #27.
  Written Problems: Section 4.5: #4 [Read example 3 first], #8b, #9a, #10, #14, #18, #TF(c)(i). Section 4.6: #4, #6, #12. Section 4.7: #6, #7b, #10a, #16, #18 [see example 9], #28 [Hint:TF(g) is true (why?)]. [Optional and Extra Credit: 4.5.#22 and 23.]
  


• Tuesday, 10/29 : Read Examples of Linear Transformations in R^2 and R^3 in Section 4.9 (you don't have to memorize these but you should be aware of them).


• Thursday, 10/31 : Read Examples 3, 4, 8 in Section 4.10.


• Homework #9 : Due Thursday, 11/7. HW#9 solutions distributed on 11/7.
  Suggested Problems: Section 4.8: #3, #7, #9, #19, #27. Section 4.9: #3, #5, #7, #9, #11, #15, #39. Section 4.10: #21, #27, #30, #TF(e)(f)(g).
  Written Problems: Section 4.8: #6, #7b and #8, #14a, #21, #30, #TF(b) [Hint: Consider A as m x n matrix and look at linear independence of rows and columns when m less than n and when m larger than n]. Section 4.9: #32b and #38 [Hint: cos(-t)=cost (t), sin(-t)= -sin(t)], #39. Section 4.10: #4 [Use matrix from Section 4.9], #20a, #24.
  


• Tuesday, 11/5 : Read the discussion of Kernel and Range of a transformation, as well as the corresponding Matrix view/ System view/ Transformation view, on Page 274 in Section 4.10.


• Thursday, 11/7 : Read Example 8 in Section 5.1.
  Read Examples 1 and 2 in Section 5.2.


• Thursday, 11/7 : [Optional Reading: Not part of the course syllabus] Read Section 5.4 and Section 5.5 to see how these ideas are applied in Differential Equations and in Markov Chains.


• Homework #10 : Due Thursday, 11/14. This is a strict deadline due to Exam#3 on Tues, 11/19. HW#10 solutions distributed in class on 11/14.
  Suggested Problems: Section 5.1: #3, #5, #13, #15, #25, #27, #33 . Section 5.2: #3, #5, #15, #19, #25, #26, #TF.
  Written Problems: Section 4.10: #21a. Section 5.1: #8, #10, #24a and #25, #34, #TF(c). Section 5.2: #8, #10, #12, #16a, #20a, TF(d)(e).
  


• Thursday, 11/14 : Read Theorem 6.1.2 and do the Example 12 on page 352 in Section 6.1.
  Read Examples 2, 3, 4, and 5 on pages 358-59 in Section 6.2.


• Thursday, 11/21 : Read Examples 7, 8, and 10 in Section 6.3.


• Homework #11 : Due Tuesday, 11/26. (Assigned on Tuesday, 11/21, and based on three lectures.) HW#11 solutions to be distributed in class on 11/26.
  Suggested Problems: Section 6.1: #2, #5, #10, #33, #35, #TF. Section 6.2: . Section 6.3: .
  Written Problems: Section 6.1: two of (#18, #20, #22), #28, #34. Section 6.2: #17, #25, #27, #37, one of (#41 OR #43), one of (#46 OR #47). Section 6.3: one of (#3b OR #4a), one of (#5 OR #10), #29, #36, #37.
  

• Tuesday, 8/20 : 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. Matrix notation and terminology, Matrix form of system of linear equations. Augmented matrix. Elementary row operations and back substitution for solving linear systems, Definitions of Row-Echelon and Reduced Row-echelon forms, examples of Row-Echelon and Reduced Row-echelon forms. (From Sections 1.1 and 1.2)
• Thursday, 8/22 : Gaussian Elimination and Gauss-Jordan algorithms - motivation. correctness and examples. Parametric form of infinite family of solutions, Identifying no solutions, 1 solution and infinitely many solutions from the augmented matrix, Leading 1s, leading and free variables. Homogenous system and its properties - trivial solution and consistency; Homogenous system with more variables then equations has non-trivial solutions (with proof). Algebra of Matrices, Equality of two matrices. (From Sections 1.2 and 1.3)
• Tuesday, 8/27 : Distribution of Course information Sheet and discussion of course organization and purpose. Algebra of Matrices, Addition and subtraction of matrices, Scalar product of matrices, Product of matrices - condition for definition, relation to dot product. (From Sections 1.2, 1.3)
• Thursday, 8/29 : Basic properties of matrix algebra, How to prove Matrix formulas/ identities/ properties. Non-commutativity of Matrix multiplication, 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. (From Sections 1.3, 1.4)
• Tuesday, 9/3 : Uniqueness of the inverse, Inverse of 2X2 matrices, Inverse of product of invertible matrices, Integer powers of a matrix, Laws of exponents for matrices, Properties of transpose, Transpose of AB, Inverse of transpose of an invertible matrix (Reading HW), Elementary matrices - relation with row operations. (From Sections 1.4, and 1.5)
• Thursday, 9/5 : Inverse of Elementary Matrix and their relation to inverse row operations, Statements equivalent to invertibility of a matrix with proofs, Method for finding inverse of a matrix and its underlying logic, Finding the inverse of a matrix, Solving linear systems with matrix inversion, Number of solutions of a system of linear equations with proof. (From Sections 1.5, 1.6)
• Tuesday, 9/10 : Number of solutions of a system of linear equations with proof, Simpler condition for invertibility of a square matrix with proof, for sq matrices AB invertible implies A and B are invertible, Two properties of solutions of non-homogenous systems equivalent to invertibility of a matrix with proofs. (From Sections 1.5, and 1.6)
• Thursday, 9/12 : 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)); Using Row operations to evaluate a determinant, Invertibility in terms of determinant, Determinant of product of matrices, Determinant of the inverse with proof. (From Sections 1.7, 2.2, and 2.3, and elsewhere)
• Tuesday, 9/17 : Invertibility in terms of determinant with proof, Determinant of product of matrices. LU decomposition of matrix-- when does it exist and how to find it using row operations in the Gaussian Elimination, Relation between L and elementary matrices and the relation between U and row echelon form, How to use the LU decomposition to easily solve a matrix equation (linear system). (From Sections 2.2, 2.3, and 9.1, and elsewhere)
• Thursday, 9/19 : Motivation and Definition of vector space, examples and non-examples of Vector Spaces, Examples (R^n, M_{m x n}, F[a,b], P_n, etc.). (From Section 4.1)
• Tuesday, 9/24 : More examples and non-examples of vector spaces (R^2 with non-standard scalar multiplication, Polynomials of degree=n, Invertible Matrices), how to prove V is a vector space, how to prove V is not a vector space - how to show an axiom is not satisfied (Axioms 4 and 5 vs. other axioms), Some elementary properties of vector spaces with proofs. (From Section 4.1)
• Thursday, 9/26 : Mid-term Exam #1.
• Tuesday, 10/1 : introduction to subspaces with examples and non-examples, Characterization of subspaces, Vector space of solution vectors of a homogenous system (Null(A)). (From Section 4.2)
• Thursday, 10/3 : Distribution and discussion of Exam#1. 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 and the smallest subspace containing S, Spanning sets for some vector spaces and subspaces. (From Section 4.2)
• Tuesday, 10/8 : Spanning sets for some vector spaces and subspaces, Conversion of a spanning set problem into a linear system problem, linear independence and its motivations, Linear independence and dependence of vectors with examples and non-examples. Discussion of EXAM #1 solutions. (From Sections 4.2 and 4.3)
• Thursday, 10/10 : Discussion of EXAM #1 solutions (concluded). 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. (From Sections 4.3, and 4.4)
• Tuesday, 10/15 : 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 with less vectors than in a basis, Dimension of a vector space, examples, dimension of the solution space of a homogenous system. (From Sections 4.4 and 4.5)
• Thursday, 10/17 : Plus/Minus theorem, How to check for basis of a vector space whose dimension is known, Converting a large spanning set or a small linearly independent set into a basis, How to extend a set of vectors into a basis for R^n, dimension of a subspace vs vector space containing it. (From Section 4.5)
• Tuesday, 10/22 : Change of basis problem and transition matrix for relating the two coordinate vectors, Relation between the two transition matrices. 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). (From Sections 4.6 and 4.7)
• Thursday, 10/24 : Mid-term Exam #2.
• Tuesday, 10/29 : Finding Basis for Row(A), Col(A) and Null(A), Using Row(A) and Col(A) to find a basis of a Euclidean subspace expressed as span(S) - the difference between the two methods, 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, Consistency theorem, Equivalent statements for rank(A) = #rows, overdetermined and underdetermined linear systems and their properties, Consistency properties of linear systems with non-square coefficient matrices. (From Sections 4.7 and 4.8)
• Thursday, 10/31 : Distribution of Exam#2 and discussion of its solutions. Extension of characterization of invertible square matrices, Linear transformations from R^n to R^m and its relation to matrix multiplication with an mxn matrix, zero transformation, Identity operator, Reflection operator as a linear operator, More examples of linear operators, 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. (From Sections 4.8, 4.9 and 4.10)
• Tuesday, 11/5 : Discussion of Exam#2 solutions (concluded). Characterization of linearity with proof using standard Euclidean basis vectors to form the standard matrix, Using the standard basis to find the standard matrix for any linear operator, Using the standard basis to find the standard matrix for any linear operator. (From Section 4.10)
• Thursday, 11/7 : 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, Invertibility and eigenvalues. Eigenvector problem and the Diagonalization problem, Definition and motivation for diagonalizability of matrices, Similar matrices, Characterization of diagonalizable matrices in terms of eigenvectors. (From Sections 5.1 and 5.2)
• Tuesday, 11/12 : 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, Geometric and algebraic multiplicities of a eigenvalue and their characterization of diagonalizability of a matrix. Inner product on a vector space, Inner product spaces, 3 different Inner products on R^n, Relation between different inner products on R^n. (From Sections 5.2, 6.1 and 6.2)
• Thursday, 11/14 : Inner products on Matrices, Polynomials, and Continuous functions, Norm and distance functions and their properties, Geometry from an i.p.s., Cauchy-Schwarz inequality, Triangle inequality, Angle between two vectors in an i.p.s., Orthogonal vectors, Generalized Pythagoras Theorem. (From Sections 6.1 and 6.2)
• Tuesday, 11/19 : Mid-term Exam #3.
• Thursday, 11/21 : 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, Coordinate vector relative to an Orthonormal basis, Projection theorem: orthogonal projection onto a subspace, Gram-Schmidt process for creating an Orthonormal basis of an inner product space with proof, QR decomposition of a matrix.(From Sections 6.2 and 6.3)
• Tuesday, 11/26 : Distribution and discussion of Mid-term Exam #3. Best approximation in an R^3 and in any ips, Best Approximation Theorem, Least squares problem, Derivation and Consistency of Normal system and usage of least square solutions to find the projection of a vector onto a subspace. Overview of Orthogonal Matrices and Orthogonal Diagonalization. (From Sections 6.4 and 7.1, 7.2)
• Thursday, 11/28 : Thanksgiving Break.

• Student companion site for the textbook
• Use/Purpose of Linear Algebra, by Oliver Knill (Harvard)
• A Self-Guided Aid to Proofs
• Linear Algebra Toolkit for practice and play.