# Math 320: Linear Algebra I David Dumas University of Illinois at Chicago Fall 2018 The JPEG compression algorithm begins with this basis for the space of 8x8 grayscale images
Textbook Friedberg, Insel, and Spence, Linear Algebra, 4ed MWF 9am in 316 Burnham Hall David Dumas MWF 10am in 503 SEO Exception: No office hours on Mon Dec 10 Charles Alley Wed 1pm-4pm in MSLC (third floor of SES) 13705

## About the final exam

The final exam will be held on Tuesday, December 11 from 10:30am to 12:30pm in 316 Burnham Hall.

The final exam will consist of a list of at least five questions, and for full credit you will be asked to complete any four of them. Two of the exam problems will posted on the web page seven days before the exam. You are encouraged to work on these questions in advance, but collaboration is prohibited.

No notes or books may be used during the final exam. No formula sheet nor list of axioms will be provided. Unless a question on the final exam specifies otherwise, the rules for writing proofs are the same as in the most recent homework assignment.

The final exam questions will be an approximately equal mix of older material (that covered by exams 1 and 2) and new material.

## Homework

Unless otherwise specified, all problems are from the textbook. Unless the book or the assignment instructs you otherwise, every problem requires you to write a proof of your answer.

It is important to write legibly if you are hand-writing solutions!

Unless specifically instructed not to for a given question, you always need to write a proof.

• Homework 0, due Wednesday August 29
• Homework 1, due Wednesday, September 5
Section 1.2 (problems begin on p12):  1 (just T/F, no proofs), 8, 11, 12, 13, 14, 16, 20, 21
• Homework 2, due Monday, September 10
Section 1.3 (problems begin on p19):  8, 10, 19, 20
Section 1.4 (problems begin on p32):  6, 10, 12
• Homework 3, due Monday, September 17
Section 1.5 (problems begin on p40):  2acde, 5, 8, 13
Section 1.6 (problems begin on p53):  2abc, 5, 8, 9
• Homework 4, due Monday, September 24
Section 1.6 (problems begin on p53):  10, 11, 21, 24
Section 2.1 (problems begin on p74):  9, 10
• Homework 5, due Monday, October 1
Section 2.1 (problems begin on p74):  13, 14, 17, 19
Section 2.2 (problems begin on p84):  2adg, 3, 4
• Homework 6, due Monday, October 8
Section 2.2 (problems begin on p84):  8, 11, 13, 16
Section 2.3 (problems begin on p96):  3, 9, 12
• Homework 7, due Monday, October 15
Section 2.4 (problems begin on p106):  2ace, 3, 4
Section 2.5 (problems begin on p116):  2bc, 3acd, 5, 6c
• Homework 8, due Monday, October 22
Section 1.3 (problems begin on p19):   26, 28, 29
Section 1.6 (problems begin on p53):   29, 35
Section 2.1 (problems begin on p74):   25, 35, 40ab
• Homework 9, due Monday, October 29 (PDF problem sheet)
• Homework 10, due Monday, November 4
Section 3.2 (problems begin on p165): 21
Section 3.3 (problems begin on p179): 4,7,8
• Homework 11, due Monday, November 12
Section 3.4 (problems begin on p194): 2adh, 5, 7, 9
Section 4.1 (problems begin on p207): 3ac, 9
• Homework 12, due Monday, November 19
Section 4.2 (problems begin on p220): 2,3,4,13,15,21,23
• Homework 13, due Monday, November 26 (PDF problem sheet)
• Homework 14, due Monday, December 3
Section 5.1 (problems begin on p256): 4abfgj, 8ab, 15a
Section 5.2 (problems begin on p279): 2be, 3bd, 12b
• Week 15: Suggested problems (not collected):
Section 5.2 (problems begin on p279): 13, 20, 22
Section 5.3 (problems begin on p308): 2ef
Section 2.6 (problems begin on p123): 3,4,6

The older assignments in the list above show only the problem numbers; previously, some of these had additional instructions or hints. If you want to see the complete original text of the older assignments, they can be found here.

• August 27: Chapter 1 and Appendix C
Skipped:
• Equation-solving algorithm from 1.4 (we'll do this in chapter 3)
• Section 1.7
• September 14: Chapter 2
• October 5: Chapter 3
• November 2: Chapter 4
• November 14: Chapter 5

## Lecture Topics

• Lecture 1 (Mon Aug 27): The definition of a vector space (Section 1.2)
• Lecture 2 (Wed Aug 29): Examples of vector spaces (Section 1.2)
• Lecture 3 (Fri Aug 31): Basic theorems in vector spaces (cancellation etc.) (Section 1.2)
• Lecture 4 (Wed Sep 5): Subspaces (Section 1.3)
• Lecture 5 (Fri Sep 7): Linear combinations and span (Section 1.4)
• Lecture 6 (Mon Sep 10): Linear independence (Section 1.5)
• Lecture 7 (Wed Sep 12): Bases (Section 1.6)
• Lecture 8 (Fri Sep 14): Replacement theorem (Section 1.6)
• Lecture 9 (Mon Sep 17): Dimension (Section 1.6)
• Lecture 10 (Wed Sep 19): Lagrange interpolation (Section 1.6)
• Lecture 11 (Fri Sep 21): Linear Transformations (Section 2.1)
• Lecture 12 (Mon Sep 24): Null space and range (Section 2.1)
• Lecture 13 (Wed Sep 26): The matrix of a linear transformation (Section 2.2)
• Lecture 14 (Fri Sep 28): Exam 1
• Lecture 15 (Mon Oct 1): The space of linear transformations (Section 2.2)
• Lecture 16 (Wed Oct 3): Composition and matrix multiplication (Section 2.3)
• Lecture 17 (Fri Oct 5): Invertibility and isomorphisms (Section 2.4)
• Lecture 18 (Mon Oct 8): Inverse matrices and applications (Section 2.4)
• Lecture 19 (Wed Oct 10): Change of coordinates (Section 2.5)
• Lecture 20 (Fri Oct 12): Change of basis (Section 2.5), sums of subspaces
• Lecture 21 (Mon Oct 15): Direct sums and projections
• Lecture 22 (Wed Oct 17): Quotient vector spaces
• Lecture 23 (Fri Oct 19): Quotient maps, first isomorphism theorem
• Lecture 24 (Mon Oct 22): Row and column operations (Section 3.1)
• Lecture 25 (Wed Oct 24): Matrix rank (Section 3.2)
• Lecture 26 (Fri Oct 26): Computing the inverse matrix (Section 3.2)
• Lecture 27 (Mon Oct 29): Systems of linear equations (Section 3.3)
• Lecture 28 (Wed Oct 31): Gaussian elimination (Section 3.4)
• Lecture 29 (Fri Nov 2): Exam 2
• Lecture 30 (Mon Nov 5): Finding a basis of the null space (Section 3.4)
• Lecture 31 (Wed Nov 7): 2x2 determinints (Section 4.1)
• Lecture 32 (Fri Nov 9): Definition of the determinant (Section 4.2)
• Lecture 33 (Mon Nov 12): Properties of the determinant (Section 4.2)
• Lecture 34 (Wed Nov 14): Row operations and the determinant (Section 4.3)
• Lecture 35 (Fri Nov 16): Reed-Solomon error-correcting codes and Lagrange interpolation
• Lecture 36 (Mon Nov 19): Introduction to eigenvalues and eigenvectors (Section 5.1)
• Lecture 37 (Wed Nov 21): Finding eigenvalues and eigenvectors (Section 5.1)
• Lecture 38 (Mon Nov 26): Diagonalizability (Sections 5.1-5.2)
• Lecture 39 (Wed Nov 28): Properties of the characteristic polynomial (Section 5.2)
• Lecture 40 (Fri Nov 30): Algebraic and geometric multiplicity (Section 5.2)
• Lecture 41 (Mon Dec 3): Necessary and sufficient conditions for diagonalizability (Section 5.2)
• Lecture 42 (Wed Dec 5): Applications of diagonalization and eigenvalues (Section 5.3)
• Lecture 43 (Fri Dec 7): Dual spaces (Section 2.6)