MATH 306 & 406 - Linear Algebra

Chapter Notes

For both Linear Algebra II and III notes.

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Chapter 1 - Vector Spaces
Chapter 2 - Finite-Dimensional Vector Spaces
Chapter 3 - Linear Maps
Chapter 4 - Polynomials (short)
Chapter 5 - Eigenvalues, Eigenvectors, and Invariant Subspaces
Chapter 6 - Inner Product Spaces

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Chapter 8 - Operators on Complex Vector Spaces
Chapter 3 (cont.) - Products and Quotients of Vector Spaces
Chapter 6 (cont.) - Finishing Inner Product Spaces
Chapter 7 - Operators on Inner Product Spaces
Chapter 9 - Operators on Real Vector Spaces

If you want to see any practice problems for this, see the following:

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HW 1 - Vector Spaces
HW 2 - Finite Dimensional Vector Spaces
HW 3 - Linear Maps
HW 4 - Inverse Maps
HW 5 - Polynomials, Invariants, and Eigenvectors
HW 6 - Finishing UT Matrices, Eigenspaces and Diagonal Matrices
HW 7 - Inner Product Spaces

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HW 1 - Products and Quotients, Duality
HW 2 - Finishing Duality, Review of Inner Product Spaces
HW 3 - Self-Adjoint and Normal Operators
HW 4 - The Spectral Theorem
HW 5 - Positive Operators and Isometries
HW 6 - Polar & Singular Value Decomposition
HW 7 - Generalized Eigenvectors, Operator Decomposition
HW 8 - Operator Decomposition, Characteristic and Minimal Polynomials
HW 9 - Jordon Form, Complexification

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Lecture 1 - Syllabus and Whatnot
Lecture 10 - Finishing LI&LD, Bases
Lecture 11 - Starting Infinite Dimensional Spaces!
Lecture 12 - Dimension
Lecture 13 - Linear Transformations
Lecture 14 - Linear Transformations ++
Lecture 15 - (Quick) Finishing our Proof for FTFLM
Lecture 16 - Types of Transformations
Lecture 17 - Continuing Matrices
Lecture 18 - Finishing Matrices, Starting Invertibility
Lecture 19 - Exam Review Day
Lecture 2 - Vector Spaces (cont.)
Lecture 20 - Invertibility (cont.)
Lecture 21 - Polynomial Spaces (4.A)
Lecture 22 (online) - Invariant Subspaces
Lecture 23 - Polynomial Operator
Lecture 24 - Finishing Eigenstuff
Lecture 25 - Eigenvalues (cont.)
Lecture 26 - Pitch on Why Diagonalization is Cool
Lecture 27 - Starting Inner Product Spaces
Lecture 28 - Continuing Inner Product Spaces
Lecture 29 - More on Orthonormality
Lecture 3 - Proofs in Linear Algebra
Lecture 30 - Practice with Gram-Schmidt
Lecture 31 - Last Lecture!
Lecture 4 - (Online) Subspaces and Constructions
Lecture 5 - Subspaces and Direct Sums
Lecture 6 - (Absent)
Lecture 7 - A Start into Chapter 2
Lecture 8 (Online) - Reference to Chapter 2
Lecture 9 - Linear Independence and Span

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Lecture 10 - Adjoint in More Detail
Lecture 11 - Finishing 7.A
Lecture 12 - More on Spectral Theorem
Lecture 1 - Intro to The Course
Lecture 14 - Starting Positive Operators and Isometries
Lecture 15 - Finishing 7C, More Applications of Linear!
Lecture 13 - Applications of Quadratic Forms
Lecture 19 - Finishing Decomposition, Some Applications
Lecture 16 - Finishing Isometries
Lecture 17 - Quadratic Forms (More Applications)
Lecture 2 - Products and Quotients of Vector Spaces
Lecture 18 - Polar Decompositions
Lecture 23 - Decomposition of an Operator
Lecture 20 - Principal Component Analysis (Application)
Lecture 22 - Finishing G. Eigenspaces, Starting 8.B
Lecture 21 - Starting Generalized Eigenvectors and Nilpotent Operators
Lecture 25 - Starting 8.C
Lecture 27 - Finishing Minimal Polynomials & Starting Jordon Block Decomposition
Lecture 24 - (finish) Decomposing Operators
Lecture 3 - Quotient Spaces
Lecture 26 - More on Minimal & Monic Polynomials
Lecture 28 - Continuing Jordan Block Decomposition
Lecture 29 - Review of Jordan-Block, Complexification
Lecture 30 - Complexification
Lecture 31 - Finishing Complexification
Lecture 6 - Matrices of Dual Maps
Lecture 7 - Ending Duality
Lecture 5 - Continuing Duality
Lecture 4 - Duality (intro)
Lecture 8 - Linear Functionals
Lecture 9 - Finishing Linear Functional Via Inner Products