**Abstract Linear Algebra**

**Math 20250 Section 55**

**Spring 2018**

**Announcements**

** **- The final exam will take place in Cobb 112. In particular it is NOT in our usual classroom!

- HW 9, although labelled as homework, is purely for practice purposes and will NOT be graded. Do not turn it in.

**General Information**

** **Instructor: Da Rong (Daren) Cheng

E-mail: chengdr "at" uchicago.edu

Office: Eckhart 409

Office hours (tentative): Tue 2pm - 3:30pm & Fri 2pm - 3:30pm

Problem session: Mon 5pm - 6pm at Eckhart 207

Class hours: MWF 12:30 - 1:20pm - Eckhart 308

Textbook: "*Linear Algebra Done Wrong" *by Treil. (Chapters 1 - 6 and part of 9.)

(Link to pdf can be found here.)

Exam dates: There will be two in-class midterms and a final exam.

1st midterm on Friday, April 20th

2nd midterm on Monday, May 14th

Final exam on Thursday, June 7th, 4pm - 6pm at Cobb 112

Grading:

Homework 10%

1st midterm 25%

2nd midterm 25%

Final 40%

**Homework **

Homework will generally be assigned every Wednesday and due the following Wednesday in class. Late homework will not be accepted under all circumstances. However, at the end of the quarter, the lowest homework score will be dropped. This means that you can miss one homework without penalty.

Homework 1 (due 4/4) [PDF][Solutions (to selected problems)]

Homework 2 (due 4/11) [PDF][Solutions (to selected problems)]

Homework 3 (due 4/18) [PDF][Solutions (to selected problems)]

Homework 4 (due 4/25) [PDF][Solutions (to selected problems)]

Homework 5 (due 5/2) [PDF][Solutions (to selected problems)]

Homework 6 (due 5/9) [PDF][Solutions (to selected problems)]

Homework 7 (due 5/16) [PDF][Solutions (to selected problems)]

Homework 8 (due 5/23) [PDF][Solutions (to selected problems)]

"Homework 9 (not graded; do not turn in)" [PDF][Solutions (to selected problems)]

**Topics covered **

This part will be updated each week to record the topics actually covered in each lecture. The numbers in parentheses (chapter followed by section) refer to the places in the textbook to which the topics roughly correspond.

**Week 1**

3/26 Vector spaces and examples (1.1); subspaces (1.7).

3/28 Examples of subspaces; consequences of the definition of vector spaces (1.1).

3/30 Linear combination; basis; linear independence. (1.2)

**Week 2**

4/2 Matrices and their algebra.

4/4 Definition and examples of linear transformations; the null space (kernel) and the range; linear transformations as a vector space.

4/6 Matrix representation of linear transformations between vector spaces; invertible linear transformations and matrices.

**Week 3**

4/9 Systems of linear equations; the reduced row echelon form and pivots; row reduction.

4/11 Invertibility criterion via row reduction.

4/13 Dimension; determining linear independence or completeness by row reduction.

**Week 4**

4/16 The rank-nullity theorem.

4/18 Sum of two subspaces; quotient spaces.

4/20 Midterm 1 [PDF][Solutions]

**Week 5**

4/23 More on isomorphisms; the first isomorphism theorem.

4/25 Determinants; existence and uniqueness; basic properties.

4/27 Determinant of a product; determinant of the transpose.

**Week 6**

4/30 Computing the determinant by row reduction; cofactor expansion.

5/2 Cofactors and the inverse; motivation for spectral theory.

5/4 Similar matrices; diagonalizability; definition of eigenvalues and eigenvectors.

**Week 7**

5/7 Example of a non-diagonalizable matrix; algebraic and geometric multiplicity; independent subspaces.

5/9 Independent subspaces; equality of algebraic and geometric multiplicity implies diagonalizability.

5/11 Definition of examples of inner product spaces; Cauchy-Schwarz inequality.

**Week 8**

5/14 Midterm 2 [Solutions]

5/16 Orthogonality; distance-minimizing property of the orthogonal projection.

5/18 Existence and uniqueness of the orthogonal projection; basic properties of the projection operator; orthogonal and orthonormal systems.

**Week 9**

5/21 The Gram-Schmidt process; orthogonal complements; the adjoint of a matrix.

5/23 The adjoint of a linear transformation; self-adjoint operators; statement of the spectral theorem for self-adjoint matrices.

5/25 Invariant subspaces; proof of the spectral theorem.