Lecture Notes for Math 115A (Linear Algebra)

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