Lecture Notes for Math 115A (Linear Algebra)

✍ Scribed by Terence Tao

• Year: 2002
• Tongue: English
• Leaves: 271
• Category: Library

✦ Synopsis

"Lecture notes from the Math 115A undergraduate course in linear algebra given in Fall 2002 at UCLA."

✦ Table of Contents

Week 1 (Sections: 1.1-1.6)
Overview of course
Overview of course
What is a vector? What is a vector space?
Definition of a vector space
(Not very important) remarks
Examples of vector spaces
Non-vector spaces
Vector arithmetic
Vector subspaces
Linear combinations
Spanning sets
Linear dependence and independence
Bases
Week 2 (Sections: 1.6-2.1)
Review of bases
Examples of bases
Rigorous treatment of bases
Dimension
Subspaces and dimension
Lagrange interpolation
Linear transformations
Week 3 (Sections: 2.1-2.3)
Review of linear transformations
Null spaces and nullity
Range and rank
The dimension theorem
Linear transformations and bases
Co-ordinate bases
The matrix representation of linear transformations
Things to do with linear transformations
Addition and multiplication of matrices
Week 4 (Sections: 2.3-2.4)
A quick review of matrices
Co-ordinate matrices and composition
Comparison between linear transformations and matrices
Matrices as linear transformations.
Invertible linear transformations
Invertible linear transformations and invertible matrices
Week 5 (Sections: 1.1-2.5)
Changing the basis
Co-ordinate change and matrices
Common sources of confusion
Week 6 (Sections: 3.1-5.1)
Review: Row and column operations on matrices
Rank of a matrix
Determinants
Geometric interpretation of determinants (optional)
Week 7 (Sections: 4.5, 5.1-5.2)
Cramer’s rule
Diagonal matrices
Eigenvalues and eigenvectors
Computing eigenvalues
Week 8 (Sections: 5.2, 6.1)
Characteristic polynomials
Tests for diagonalizability
Inner product spaces
Inner products and length
Week 9 (Sections: 6.1-6.2)
Orthogonality
Orthonormal bases
The Gram-Schmidt orthogonalization process.
Orthogonal complements
Week 10 (Sections: 3.1-5.1)
Linear functionals
Adjoints
Normal operators
Self-adjoint operators
Assignment 1 (Sections: 1.1-1.6)
Assignment 2 (Sections: 1.6-2.1)
Assignment 3 (Sections: 2.1-2.3)
Assignment 4 (Sections: 2.3-2.4)
Midterm
Assignment 5 (Sections: 2.4-2.5)
Assignment 6 (Sections: 3.1-3.2, 4.1-4.4)
Assignment 7 (Sections: 4.4, 5.1-5.2)
Assignment 8 (Sections: 5.2, 6.2-6.3)
Final Examination