Linear Algebra: Schedule

We discussed the basic terminology and concepts necessary for understanding linear algebra. This includes some set theory notation and a brief introduction to complex numbers.

As a first quiz (due Sunday 2020-02-14 before 11:59pm), follow these instructions (printer-friendly) to familiarize yourself with Gradescope submissions.

An introduction to solving linear systems via the method of Gaussian elimination.

Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon One.I.1 – One.I.2 and Matthews 1.2.

Using augmented matrices as a tool for representing and solving linear systems. Solution sets and the possible types of solutions to linear systems.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon One.I.2 and One.III.1 and Matthews 1.3 – 1.4.

Matrix operations and matrix equations. Some discussion of homogeneous and inhomogeneous linear systems.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon One.I.3 and Three.IV.1 – Three.IV.3 and Matthews 1.5 and 2.1.

Geometric objects in linear algebra. Vector addition, the dot product, and properties.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon One.II.1 – One.II.2.

The dot product and geometry. Algebraic properties of vector addition. The Cauchy-Schwarz Inequality. The Triangle Inequality. Proved a formula relating the dot product to geometry.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon One.II.2.

Reduced row echelon form. The Linear Combination Lemma. Row equivalence of matrices.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon One.III.1 – One.III.2.

More on reduced row echelon form and row equivalence.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon One.III.2.

Linear transformations arising from matrices.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Three.II.1 (up to Example 1.8).

This meeting was a review session for exam one; students must bring questions.

I linked some practice exams from previous semesters below. I do NOT claim that these practice exams are representative of our exam in any way–in fact, I haven't looked at them at all. Use them at your own risk!

First Sample Exam (Solutions), Second Sample Exam (Solutions), Third Sample Exam (Solutions)

Exam one took place during the normal lecture time on Zoom.

Definition and examples of vector spaces.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Two.I.1.

A vector space within a vector space… It's spaception….

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Two.I.2.

Span of a set of vectors in a vector space.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Two.I.2.

Introduction to and properties of linear independence.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Two.II.1.

Introduction to bases of a vector space (with an emphasis on computations).

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Two.III.1.

More on bases and dimension (with an emphasis on theory).

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Two.III.2.

Some final words on bases and dimension, including several computations.

Introduction to linear maps of arbitrary vector spaces.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Three.II.1.

Range and kernel of linear maps. Injective and surjective linear maps.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Three.II.2.

The rank-nullity formula (with proof). More simple propositions about linear maps.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Three.I.1-2 and Three.II.1.

Column and row spaces of a matrix, matrix rank, and consequences.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Two.III.3.

Linear maps from a vector space to itself. Linear maps \(\mathbb{R}^n \to \mathbb{R}^n\) are determined by square matrices (note: something more general is true…). Computing the inverse of a linear automorphism via matrix inversion (this is why we think about operators on \(\mathbb{R}^n\)).

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Three.II.1 and Three.IV.4.

Quiz 01 (printer-friendly) is due via Gradescope by 11:59pm tonight!

This meeting was a review session for exam two; students must bring questions.

I linked some practice exams from previous semesters below. I do NOT claim that these practice exams are representative of our exam in any way–in fact, I haven't looked at them at all. Use them at your own risk!

First Sample Exam (Solutions), Second Sample Exam (Solutions), More Practice (Solutions)

Exam two took place during the normal lecture time on Zoom.

Row reduction and elementary matrices. Matrix inversion and elementary matrices.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Three.IV.3 and Three.IV.4.

Introduction to the determinant of a matrix transformation. Some properties of the determinant.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Four.I.1 – Four.I.2 (note: Hefferon uses vertical bars around a matrix to denote its determinant).

More on properties of the determinant.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Four.I.1 – Four.I.4.

Introduction to changes of coordinate.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Three.V.1 – Three.V.2.

Quiz 02 (printer-friendly) is due via Gradescope by 11:59pm tonight!

Review of basis change (huh, that picture must be important: Chris keeps drawing it). Introduction to the eigenvectors, eigenvalues, and characteristic polynomial of a linear operator (mostly we did some simple computations).

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Five.II.3.

Homework: Read Hefferon Five.I (it's only 3.5 pages).

Review of the complex numbers (see Hefferon Five.I for a deeper treatment). Can you find the mistake? :)

More on the computation of eigenvectors and eigenvalues, as well as a short introduction to similarity of matrices.

Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Five.II.1 – Five.II.3 and my notes (printer-friendly).

Computation of eigenspaces and similarity of matrices.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Five.II.1 – Five.II.3 and my notes (printer-friendly).

Quiz 03 (printer-friendly) is due via Gradescope by 11:59pm tonight!

Diagonalization of matrices (hence linear operators).

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Five.II.1 – Five.II.3 and my notes on eigenspaces (printer-friendly) and diagonalization (printer-friendly).

Quiz 04 (printer-friendly) is due via Gradescope by 11:59pm tonight!

More on diagonalization of matrices (hence linear operators), with a focus on computations.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Five.II.1 – Five.II.3 and my notes on eigenspaces (printer-friendly) and diagonalization (printer-friendly).

Quiz 05 (printer-friendly) is due via Gradescope by 11:59pm tonight!

Orthogonal projection. Orthogonal bases of subspaces of \(\mathbb{R}^n\) and the Gram-Schmidt process.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Three.VI.1 – Three.VI.2.

Quiz 06 (printer-friendly) is due via Gradescope by 11:59pm tonight!

Brief introduction to orthogonal matrices and orthonormal bases in \(\mathbb{R}^n\). Orthogonal complements of subspaces of \(\mathbb{R}^n\).

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Hefferon Three.VI.2 – Three.VI.3.

Quiz 07 (printer-friendly) is due via Gradescope by 11:59pm tonight!

This meeting was a review session for exam three; students must bring questions.

I linked some practice exams from previous semesters below. I do NOT claim that these practice exams are representative of our exam in any way–in fact, I haven't looked at them at all. Use them at your own risk!

First Sample Exam (Solutions), Second Sample Exam (Solutions), Third Sample Exam (Solutions)

Quiz 08 (printer-friendly) is due via Gradescope by 11:59pm tonight!

Note: Regrade requests for quizzes 2 through 6 due tonight.

Exam three took place during the normal lecture time on Zoom.

A gentle introduction to symmetric matrices.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly). Further Reading: Beezer TSM and MCC.

A proof that the eigenvalues of real symmetric matrices are all real. Some simple consequences of this fact.

Lecture notes (printer-friendly). Practice problems (printer-friendly).

More on orthogonal diagonalization of symmetric matrices.

Recording. Lecture notes (printer-friendly). Practice problems (printer-friendly).

I held an (optional but encouraged) review session on 21 May 2021 (PDF, printer-friendly, partial recording).

I linked some practice exams from previous semesters below. I do NOT claim that these practice exams are representative of our exam in any way–in fact, I haven't looked at them at all. Use them at your own risk!

First Sample Exam (Solutions), Second Sample Exam (Solutions), Third Sample Exam (Solutions), Fourth Sample Exam (Solutions), Fifth Sample Exam (Solutions)

I held my last open office hours for this course on 24 May 2021.

Our final exam took place Tuesday, 25 May 2021 from 8:00am – 10:00am in the usual Zoom room.