**Instructor:** Hemanshu Kaul

**Office:** 125C, Engineering 1

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

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

**Time:** 11:25am, Monday and Wednesday.

**Place:** 204, Siegel Hall.

**Office Hours:** 1:45pm-3pm Monday, and 4:30pm-5:30pm Wednesday, 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 syllabus for 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, 8/29*: All the Exam dates have been announced below.*Monday, 8/22*: Check this webpage regularly for homework assignments, announcements, etc.

*Exam #1*: Monday, September 26th. Topics: All the topics corresponding to the HW#1, HW#2, HW#3, HW#4.*Exam #2*: Monday, October 31st. Topics: All the topics corresponding to the HW#5, HW#6, HW#7, HW#8.*Exam #3*: Wednesday, November 30th. Topics: All the topics corresponding to the HW#9, HW#10, HW#11.*Final Exam*: Tuesday, December 6th, 8am to 10am. Topics: All topics studied during the semester, except those studied on Monday, Nov 28th.

You only have to submit solutions to

However, solving a majority of the

Be sure to staple the pages together and write your name (and that of any collaborator), course number, assignment number, and the date of submission on the front.

*Wednesday, 8/24*: Read Examples for Row-Echelon and Reduced Row-echelon forms; definitions of free and leading variables; and theorems 1.2.1 and 1.2.2 from Section 1.2.

Due Wednesday, 8/31. HW#1 solutions distributed in class on 9/7.__Homework #1__:

Suggested Problems: Section 1.1: #4, #7, #11. Section 1.2: #1, #3, #5, #14, #18, #27, #36, True/False Exercises #(g), #(i).

Written Problems: Section 1.1: #5, True/False Exercises #(c). Section 1.2: #2aeg, #4ac, #7, #13, #16, #20, #25, #34.

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

Find A and B such that AB is not equal to BA.Due__Homework #2__:~~Wednesday, 9/7~~Thursday, 9/8, 1pm in my mailbox at 210, E1. HW#2 solutions distributed in class on 9/19.

Suggested Problems: Section 1.3: #1, #3, #5, #7, #13, #20b, #28. Section 1.4: #1, #2.

Written Problems: Section 1.3: #1abc, #5abc, #8bf, #12a, #16, #18, #20a, #24bc, #27, True/False Exercises #(m), (o). Section 1.4: #44, True/False Exercises #(j).

*Wednesday, 9/7*: Read Examples #4 and #5 in section 1.5 (pg. 55-57).Due Wednesday, 9/14. HW#3 solutions distributed in class on 9/19.__Homework #3__:

Suggested Problems: Section 1.3: #2, #6. Section 1.4: #3, #4, #5, #11, #16, #22, #27, #29, #34, True/False Exercises #(b)(c). Section 1.5: #4, #5, #7, #20.

Written Problems: Section 1.3: #5gh. Section 1.4: #8, #17,~~#26~~#25, #28, #31, #51. Section 1.5: #6a, #8c, #17, #39.

*Monday, 9/12*: 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 68.Due Wednesday, 9/21. This is a strict deadline as Exam#1 is on next Monday. HW#4 solutions distributed in class on 9/21.__Homework #4__:

Suggested Problems: Section 1.6: #5, #15, #19, T/F#(f). Section 1.7: #10, #17, #18, #23, #25, #33, #37, #42. Section 2.1: #17, #10, #23, #24. Section 2.2: #10, #17, #21, #26, T/F#(e). Section 2.3: #7, #15, #35.

Written Problems: Section 1.5: #30 and #34, #42. Section 1.6: #8, #16, #18a, #22, T/F#(b). Section 1.7: #24, #28, #32b. Section 2.1: #16. Section 2.2: #11, #23, #31. Section 2.3: #35d, #38.*Wednesday, 9/14*: Do Examples 3-5 on page 96-97 to practice co-factor expansion for calculating determinant.

Do example 5 on page 104 for a combination of Row operations and Cofactor expansion.*Monday, 9/19*: Do Example 3 on page 483-484 for an example of direct construction of L from the Gaussian elimination procedure.

Do Example 2 on page 74-75 for an example of network flows.Due Wednesday, 10/5. HW#5 solutions distributed in class on 10/12.__Homework #5__:

Suggested Problems: Section 9.1: #2, #5, #14, #19. Section 1.8: #2, #7. Section 3.1: #5, 7, 10, 14, 22. Section 3.2: #2, #7, #17, #27, T/F #(g)#(h). Section 4.1: #3, #6, #9, #11, #23, #25.

Written Problems: Section 9.1: #6, #16. Section 1.8: #4. Section 3.1: #20, #29, #30. Section 3.2: #6c, #22, T/F #(j). Section 4.1: #2, #4, #8, #27, and these two problems .*Wednesday, 9/28*: Read Examples 1 and 8 in Section 4.1.*Wednesday, 10/5*: Read Examples 15-16 on pages 186-187. Read Theorem 4.2.5 on page 188.Due Wednesday, 10/12. HW#6 solutions distributed in class on 10/17.__Homework #6__:

Suggested Problems: Section 4.2: #1, #2, #3, #4, #5, #8, #9, #10, #11, #12, #17, #19, #20, T/F(abdeghk).

Written Problems: Section 4.2: #1cd, #2beg, #3bc, #4ab, #8ac, #9a, #10a, #11c, #13, #18.

*Wednesday, 10/12*: Read the proof of Theorem 4.3.3 on page 196. Read examples 5 and 6 on page 204.Due Wednesday, 10/19. HW#7 solutions distributed in class on 10/24.__Homework #7__:

Suggested Problems: Section 4.3: #3b, #4a, #7, #10, #12, #15, #16, #18, T/F#abf. Section 4.4: #3a, #4c.

Written Problems: Section 4.3: #3a, #4d, #9, #11, #13, #14. Section 4.4: #3b, #4a, #5.

*Wednesday, 10/19*: Read examples 6, 7, 8, 9 on page 230-234 (these will be discussed in class on Monday).Due Wednesday, 10/26. This is a strict deadline as Exam#2 is on next Monday. HW#8 solutions distributed in class on 10/26.__Homework #8__:

Suggested Problems: Section 4.4: #9, #10, #11, #14. Section 4.5: #1, #8, #9, #10, #12, #13, #19, #20. Section 4.7: #3, #5, #6, #7, #9, #13.

Written Problems: Section 4.4: #9a, #10b, #12. Section 4.5: #4, #8b, #9a, #11, #12a, #14. Section 4.7: #4, #5b, #6c, #7b, #9d. [Optional and Extra Credit: 4.5.#18 and Chapter 4 Supplementary Exercise#7.]

Due Wednesday, 11/9. Note this HW is longer than usual as its based on 3 lectures. Get started on it right away. HW#9 solutions distributed in class on 11/14.__Homework #9__:

Suggested Problems: Section 4.7: #7b, #9d, #11a, T/F exercises. Section 4.8: #5, #7, #9, #14, #15. Section 4.9: #1, #3, #5, #8, #9, #12, #15, #19. Section 4.10: #5, #7, #9, #13, #16ab, #26, #27.

Written Problems: Section 4.7: #12b, #13, #19. Section 4.8: #6, #10, #12b. Section 4.9: #6acd, #10c. Section 4.10: #6b, #8b, #14a, #18, #23, #24ab, #25a, #29.

*Monday, 11/7*: Read Example 3 on page 221. Read Table 1 on page 306.Due Wednesday, 11/16. HW#10 solutions distributed in class on 11/21.__Homework #10__:

Suggested Problems: Section 4.6: #5, #7, #14, #21, #23, T/F exercises. Section 5.1: #1, #5, #14, #24, #26ab. Section 5.2: #2, #7, #9, #12, #18, #22, #31.

Written Problems: Section 4.6: #6, #10, #24. Section 5.1: #8a, #13, #16a, #23, #27a, #28. Section 5.2: #4, #14, #16, #24, #28.

Due MONDAY, 11/28. This is a strict deadline as Exam#3 is on Wednesday, 11/30. HW#11 solutions distributed in class on 11/28.__Homework #11__:

Note this HW is longer than usual as its based on 3 lectures. Get started on it right away and ask for help early.

Suggested Problems: Section 6.1: #3, #4, #7, #8, #11, #13, #16, #25, T/F exercises. Section 6.2: #3, #5, #8, #11, #19, #23, #26. Section 6.3: #5, #6, #10, #12, #16, #19, #21, #22a, #31.

Written Problems: Section 6.1: #10, #24bc, #26, #27. Section 6.2: #7, #13, #16c, #17, #21, one of (#24 OR #25). Section 6.3: #5b, #7b, one of (#10b OR #12a), #16b, #25, #28, #29bf.

*Monday, 8/22*: Discussion of course organization and purpose.**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, 8/24*: Distribution of Course information Sheet and discussion of course organization and purpose; Course survey. Definitions of Row-Echelon and Reduced Row-echelon forms, examples of Row-Echelon and Reduced Row-echelon forms, parametric form of infinite family of solutions, Identifying no solutions, 1 solution and infinitely many solutions from the augmented matrix,**Gaussian Elimination and Gauss-Jordan algorithms**. (From Section 1.2 and elsewhere)*Monday, 8/29*: Leading 1s,**leading and free variables**, 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, Scalar product of matrices, Product of matrices - condition for definition, relation to dot product. (From Sections 1.2 and 1.3)*Wednesday, 9/2*: Product of matrices - 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, Basic**properties of matrix algebra**, 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)*Monday, 9/5*: Labor Day Holiday.*Wednesday, 9/7*: 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,**Elementary matrices**- relation with row operations, Inverse of Elementary Matrix and their relation to inverse row operations, Method for finding inverse of a matrix and its underlying logic. (From Sections 1.3, 1.4, and 1.5)*Monday, 9/12*:**Finding the inverse of a matrix**, Three statements equivalent to invertibility of a matrix with proofs,**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, 9/14*: 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)), Invertibility in terms of determinant with proof, Determinant of product of matrices, Determinant of the inverse with proof, Cofactor of an entry, Cofactor expansion for finding determinant. (From Sections 2.1, 2.2, and 2.3, and elsewhere)*Monday, 9/19*:**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), An application of linear systems to network flows. (From Sections 9.1, 1.8, and elsewhere)*Wednesday, 9/21*: An application of linear systems to electric circuits using Ohm's and Kirchoff's laws,**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 Sections 1.8, 3.1, and 3.2)*Monday, 9/26*: Mid-term Exam #1.*Wednesday, 9/28*: Definition of**vector space**, examples and non-examples of Vector Spaces, 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, 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)*Monday, 10/3*: Distribution of Exam#1 and discussions of its solutions. Some elementary properties of vector spaces with proofs, introduction to**subspaces**with examples and non-examples. (From Section 4.2)*Wednesday, 10/5*: 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 4.2)*Monday, 10/10*: Fall Break*Wednesday, 10/12*: Span(S) is a subspace and the smallest subspace containing S, 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, 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. (From Sections 4.2, 4.3, and 4.4)*Monday, 10/17*: 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 vector space, examples, dimension of the solution space of a homogenous system,**Plus/Minus theorem**, How to check for basis of a vector space whose dimension is known. (From Sections 4.4 and 4.5)*Wednesday, 10/19*: 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**, 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 Sections 4.5 and 4.7)*Monday, 10/24*: 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 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. (From Sections 4.7 and 4.8)*Wednesday, 10/26*: 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, 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. (From Sections 4.8 and 4.9)*Monday, 10/31*: Mid-term Exam #2.*Wednesday, 11/2*: 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,**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. (From Sections 4.9 and 4.10)*Monday, 11/7*: Using the standard basis to find the standard matrix for any linear operator,**Change of basis problem and transition matrix**for relating the two coordinate vectors, Relation between the two transition matrices,**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, motivation for diagonalizability of matrices: Positive integral powers of a matrix. (From Sections 4.10, 4.6, 5.1 and 5.2.)*Wednesday, 11/9*: Distribution of Exam#2 and discussions of its solutions. Definition and motivation for diagonalizability of matrices,**Similar 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, Geometric and algebraic multiplicities of a eigenvalue and their characterization of diagonalizability of a matrix. (From Section 5.2)*Monday, 11/14*: Conclusion of the discussion of the solutions of Exam#2,**Inner product on a vector space**, Inner product spaces, 3 different Inner products on**R**^n, Relation between different inner products on**R**^n, Inner products on Matrices, Polynomials, and Continuous functions,**Norm and distance**functions and their properties, Unit-circles/spheres in ips,**Cauchy-Schwarz inequality**with proof. (From Sections 6.1 and 6.2)*Wednesday, 11/16*: Cauchy-Schwarz inequality, Triangle inequality with proof, Angle between two vectors in an i.p.s., Orthogonal vectors, 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**, Coordinate vector relative to an Orthonormal basis with proof, Normalization of an Orthogonal set, Norm, distance, and inner product using an orthonormal basis, Linear independence of orthogonal sets with proof,**Projection theorem: orthogonal projection onto a subspace**. (From Sections 6.2 and 6.3)*Monday, 11/21*: Projection theorem, Orthogonal projection formulas using orthogonal and orthonormal bases,**Gram-Schmidt process**for creating an Orthonormal basis of an inner product space with proof,**QR decomposition**problem and solution using G-S process. (From Section 6.3)*Wednesday, 11/23*: Thanksgiving Break.*Monday, 11/28*: 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 (requires orthogonal-complement( Row(A)) = Null(A)), Consistency of**Normal system**and usage of least square solutions to find the projection of a vector onto a subspace. (From Section 6.4)*Wednesday, 11/30*: Mid-term Exam #3.