**Instructor:** David Nadler

**Office Hours:** Thursdays 12:30-2:30pm, 815 Evans Hall, or by appointment.

Office hours before final: Tuesday, December 4, 2:10-3:00pm, 815 Evans; Thursday, December 6, 12:00-2:00pm, 891 Evans.

**GSIs**

- Alex Hening, Office hours:
Wednesdays 1:00-3:00pm, 937 Evans Hall.

Extra office hours before first midterm: Monday, September 17, 3:30-4:30pm. - James McIvor, Office hours: Mondays 3:00-5:00pm and Wednesdays 4:30-5:30pm, 1062 Evans Hall.

Extra office hours before final: Friday, December 7, 2:00-4:00pm. - Dan Sparks, Office hours:
Tuesdays 12:00-1:00pm, Wednesdays 9:00-10:00am, Thursdays 2:30-3:30pm, 1060 Evans Hall.

Extra office hours before final: Monday, December 3 and Tuesday, December 4, 10:00-12:00am. - James Tener, Office hours: Tuesdays and Thursdays 3:00-4:00pm, 935 Evans Hall.

Extra office hours before first midterm: Monday, September 17, 3:00-5:00pm.

Extra office hours before final: Monday, December 10, 3:00-4:00pm.

**Lectures:** Tuesdays and Thursdays 9:30-11am, 105 Stanley Hall

**Discussion sections:** Wednesdays,
see Times and Places

**Course Control Number:** 54184

**Prerequisites:** Math 54 or equivalent preparation in linear algebra.

**Required text:**
Friedberg, Insel, Spence,
*Linear Algebra*, Pearson, 4th edition (2003).

**Grading policy:** based on homework (20%), two midterm exams
(25% for higher of two scores; 15% for lower of two scores), and final exam (40%).

Two midterm exams during lecture meeting:

- Midterm 1:
**Tuesday, September 18, 2012**, Material: Sections 1.2-1.6 (with the exception of Lagrange Interpolation) and Appendix C. - Midterm 2:
**Tuesday, October 16, 2012**, Material: Chapters 2-3.

Final Exam: **Tuesday, December 11, 2012, 3-6pm (Exam group 7)**

Academic honesty: You are expected to rely on your own knowledge and ability, and not use unauthorized materials or represent the work of others as your own.

There will be no make-up midterms or final exams. No late homework will be accepted.

Grades of Incomplete will be granted only for dire medical or personal emergencies that cause you to miss the final, and only if your work up to that point has been satisfactory.

Homework is due **Fridays by 3pm** at your GSI's office
or mailbox. **Please follow your individual GSI's instructions as to
where to turn it in.**

You are encouraged to discuss ideas with other students. However, you must write and hand in your solutions independently.

Each week, two selected problems from the homework assignment will be graded. Solutions to all problems will be posted.

When calculating grades, we will drop your two lowest homework scores and use only your remaining scores.

- Due
**Friday, August 31, 2012**.

Section 1.2: problems 6, 16, 21.

Section 1.3: problems 8, 9, 19, 23, 29, 30.

Section 1.4: problems 5, 15.

Additional problem: Prove that every field F contains either the field of rational numbers Q or a finite field F_p = Z/pZ for a prime p. - Due
**Friday, September 7, 2012**.

Section 1.5: problems 3, 9, 11, 13.

Section 1.6: problems 4, 9, 13, 17, 20, 26, 29, 34.

- Due
**Friday, September 14, 2012**.

Section 2.1: problems 6, 11, 12, 20, 22, 24, 35.

Section 2.2: problems 2b, 6, 8, 12, 16.

- Due
**Friday, September 21, 2012**.

Section 2.3: problems 2, 12, 13. - Due
**Friday, September 28, 2012**.

Section 2.4: problems 2, 3, 5, 6.

Section 2.5: problems 2, 7.

Section 2.6: problems 3, 9, 19, 20.

- Due
**Friday, October 5, 2012**.

Section 3.1: 12.

Section 3.2: 3, 4, 5 (a), (b), (c), 6 (a), (b), (c), 7, 14, 17.

- Due
**Friday, October 12, 2012**.

Section 3.3: 2 (a), (b), 3 (a), (b), 8, 10.

Section 3.4: 2 (a), (f), 12.

- Due
**Friday, October 19, 2012**.

Section 4.3: 15, 21. - Due
**Friday, October 26, 2012**.

Section 5.1: 3 (a), 5, 8, 9, 11, 12, 14, 20.

- Due
**Friday, November 2, 2012**.

Section 5.2: 3 (f), 7, 12, 13, 18.

Section 5.4: 3, 6 (a), 18, 19, 42.

- Due
**Friday, November 9, 2012**.

Section 6.1: 4 (a), 8 (a), 9, 11, 12.

Section 6.2: 2 (b), 6, 13, 19 (b).

- Due
**Friday, November 16, 2012**.

Section 6.3: 2 (b), 7, 8, 12.

Section 6.4: 2 (a), 3, 6, 9.

- Due
**Friday, November 30, 2012**.

Section 7.1: 1, 3 (a), 13.

Section 7.2: 1, 4 (a), 5 (a), 6.

Section 7.3: 1, 2 (d), 5.

- Some suggested problems for Sections 6.5, 6.6 not to be collected.

Section 6.5: 6, 11, 17.

Section 6.6: 2, 4, 5.

- Vector spaces.
- Subspaces. Intersection, sum. Direct sum.
- Span, linear independence, and bases.
- Dimension of a finite-dimensional vector space.
- Linear maps.
- Nullspace, range and rank of a linear map.
- Matrix of a linear map.
- Invertible linear maps.
- Eigenvalues and eigenvectors.
- Inner product spaces.
- Orthonormal bases and the Gram-Schmidt procedure.
- Orthogonal projections; applications.
- Adjoints.
- Self-adjoint and normal operators; spectral theorem.
- Operators on complex vector spaces.
- Characteristic polynomials and minimal polynomial.
- Jordan form.

Key concepts: field axioms; basic properties of fields; examples: rational, real, and complex numbers; examples and non-examples of finite fields; characteristic of a field; vector space axioms.

Reading: Section 1.2, Appendix C.

Key concepts: row vectors, column vectors, matrices; functions, polynomials; subspace characterizations; intersection of subspaces; transpose of matrix, symmetric and skew- or anti-symmetric matrices; trace of matrices, diagonal matrices; upper/lower triangular matrices.

Reading: Section 1.2, 1.3.

Key concepts: sum of subspaces; direct sum of vector spaces; linear combinations; span of a subset; generating sets.

Reading: Section 1.4, 1.5.

Key concepts: linear dependence and independence; bases.

Reading: Section 1.5, 1.6.

Key concepts: finite bases; constructing generating sets; finding bases; Replacement Theorem; dimension.

Reading: Section 1.6.

Key concepts: maps of sets; linear maps of vector spaces; kernels and images; nullity and rank, Dimension Theorem.

Reading: Section 2.1.

Key concepts: injective, surjective, bijective maps of sets; isomorphisms of vector spaces; coordinates with respect to a basis; matrices with respect to bases; the vector space of linear transformations.

Reading: Section 2.1, 2.2.

Key concepts: compositions of maps; basic properties of compositions; multiplication of matrices; isomorphisms and inverses; every finite dimensional vector space is isomorphic to coordinate space.

Reading: Section 2.3, 2.4.

Key concepts: change of coordinate matrices; similar matrices; dual spaces and dual bases.

Reading: Section 2.5, 2.6.

Key concepts: elementary matrices and operations; rank of matrix; simplifying matrices; calculating inverses.

Reading: Section 3.1, 3.2.

Key concepts: consistent and inconsistent systems; homogenous and nonhomogenous systems; relation of solutions and null space; criteria for existence of solutions; Gaussian elimination and reduced row echelon form; solving systems of linear equations in reduced row echelon form.

Reading: Section 3.3, 3.4.

Key concepts: dual spaces; dual bases; transposes.

Reading: Section 2.6.

Key concepts: geometry of determinants of 2x2-matrices; inductive definition of determinants in general; linearity with respect to fixed row; effect of row operations.

Reading: Section 4.1, 4.2.

Key concepts: determinant of product of matrices is product of determinants; determinant is nonzero if and only if matrix is invertible; determinants of elementary matrices; determinant of transpose; effect of column operations.

Reading: Section 4.3.

Key concepts: eigenvectors and eigenvalues; diagonalizability; characteristic polynomial; eigenvalues equal roots of characteristic polynomial.

Reading: Section 5.1.

Key concepts: algorithm to check diagonalizability; split or non-split polynomials; multiplicity of roots of characteristic polynomial; dimension of eigenspace.

Reading: Section 5.2.

Key concepts: examples of all possible outcomes of algorithm; comparison of multiplicity of eigenvalue to dimension of eigenspace; applications of diagonalization.

Reading: Section 5.2.

Key concepts: invariant subspace; generating vector; characteristic polynomial of block upper triangular matrix; Cayley-Hamilton Theorem.

Reading: Section 5.4.

Key concepts: review of complex numbers; Fundamental Theorem of Algebra; definition of inner product; inner products from matrices.

Reading: Section 6.1, Appendix D.

Key concepts: basic inequalities; orthogonal sets; unit vectors; orthonormal sets.

Reading: Section 6.1.

Key concepts: Gram-Schmidt process; orthonormal bases; coefficients with respect to orthonormal bases; orthogonal complements; direct sum of subspace and its orthogonal complement.

Reading: Section 6.2.

Key concepts: orthogonal projections; existence and uniqueness of adjoint of linear transformation; compatibility with conjugate-transpose of a matrix.

Reading: Section 6.3

Key concepts: relation of eigenvalues for transformation and its adjoint; Schur's theorem; Spectral theorem; inner products are self-adjoint matrices with positive eigenvalues; orthogonal and unitary transformations.

Reading: Section 6.4, 6.5, 6.6.

Key concepts: definition of Jordan canonical form; notions of generalized eigenvector and eigenspace; dot pictures organizing Jordan canonical form data.

Reading: Section 7.1, 7.2.

Key concepts: decomposition into generalized eigenspaces; cycles of generalized eigenvectors; minimal polynomial.

Reading: Section 7.1, 7.2, 7.3

Some previous Math 110 course home pages:

- Mark Haiman's Spring 2012 Math 110.
- Ming Gu's Fall 2011 Math 110.
- Olga Holtz's Fall 2006 Math 110.

George Bergman's notes:

- Supplemental Exercises to Friedberg, Insel and Spence's "Linear Algebra", 4th Edition
- Some notes on sets, logic, and mathematical language

- Strang's Spring 2010 MIT course and Fall 2011 MIT course

- Axler,
*Linear Algebra Done Right* - Halmos,
*Finite Dimensional Vector Spaces* - Halmos,
*Linear Algebra Problem Book*