**UM 102: Undergraduate Calculus and Linear Algebra, last taught by me in the January Semester of 2015. I shall teach it again in the January semester of 2020.**

**My office hours for this course are 5:30 PM to 6:30 PM on Tuesdays and Thursdays. The problem sets of 2015 are found below.**

Exercise Set I: Click here for the PDF file.

Exercise Set II: Click here for the PDF file.

Exercise Set III: Click here for the PDF file. It says Problem Set V because it was the fifth problem set in 2014. It is Problem Set III for 2015.

Mid Sem Question paper is here.

Exercise Set IV: Click here for the PDF file. It says Problem set III because it was the third problem set in 2014. It is problem set IV for 2015.

**First class**: Revision of basics of vector spaces and Rank-Nullity Theorem. Reference: Apostol's Calculus volume I, section 16.3.

**Second class**: Inner products and Cauchy-Schwarz inequality. Reference: Apostol's Calculus volume I, section 15.10.

**Third class**: Orthogonality, Parseval's formula, Gram-Schmidt process. Reference: Apostol's Calculus volume I, section 15.11 and 15.13.

**Fourth class**: Gram-Schmidt continued, uniqueness. Projections. Reference: Apostol's Calculus volume I, section 15.14.
Documents: Legendre Polynomials notes from a site at Rochester and closer home a very good set of notes from SERC.

**Fifth class**: Approximation theorem, system of equations. Reference: Apostol's Calculus volume I, section 15.15 and 16.17.

**Sixth class**: Determinants - motivation through cross product, axiomatic definition. Reference: Apostol's Calculus volume II, most of Chapter 3.

**Seventh class**: Existence and uniqueness of the determinant function, computations. Some elementary examples. Reference: Apostol's Calculus volume II, most of Chapter 3.

**Eighth class**: Hermitian operators on an inner product space - orthogonality of eigenvectors corresponding to different eigenvalues, orthonormal eigenbasis. Reference: Apostol's Calculus volume II, first half of Chapter 5.

**Ninth class**: Diagonalization of hermitian matrices. Reference: Horn and Johnson, Matrix Analysis, second edition, section 2.3.1.

**Tenth class**: Limit and continuity. Reference: Apostol's Calculus volume II, Section 8.4. Also see Thomas' calculus, section 14.2

**Eleventh and Twelfth class**: Differentiation in n-dimensial real space. Directional, partial and total derivatives. Existence of total derivative implies continuity. Example that the existence of all directional derivatives does not imply continuity in general. Reference: Apostol's Calculus volume II, Section 8.6 to 8.12.

**Thirteenth class**: Quiz 1.

**Fourteenth class**: A sufficient condition for differentiability. Reference: Apostol's Calculus volume II, Section 8.13

**Fifteenth class**: Chain rule for differentiation - Jacobian. Reference: Apostol's Calculus volume II, Section 8.15

**Sixteenth class**:

**Seventeenth class**:

**Eighteenth class**:

**Nineteenth class**:

**Twentieth class**:

**Twenty-first class**:

**Twenty-second class**:

**Twenty-third class**: Ordinary differential equation - first order non-homogeneous.

I taught UM 102 in 2014 too. The course page for that is here. The mid sem paper for 2014 is here.