Linear Algebra, Math 405 Fall 2008
Course Information
 PREREQUISITES: MATH 240 or MATH 461
 Course Description: This course contains an abstract
treatment of finite dimensional vector spaces, linear transformations and their
invariants.

Contents
Vector spaces
Vector spaces and subspaces
Linear combinations, span and linear (in)dependence
Note: On the transitivity of linear spans [
pdf file ]
Assignment #1 [
pdf file ] ... with answers [ pdf file]
Bases and dimension
Row equivalence in R^{N} and rank
Coordinates
Note: An exchange theorem and the notion of dimension [
pdf file ]
Assignment #2 [ pdf file ] ... with answers [ pdf file]
Linear transformations
Linear transformations (homomorphism, isomorphism, ...)
Vector spaces of finite and infinite type; Rank and nullity, Sylvester theorem
(*) Direct sums and products of vector spaces
Assignment #3 [ pdf file ] ... with answers [ pdf file]
Matrix representation of linear transformations
Sum and product of matrices; change of bases
Note: On matrix representation of linear transformations [
pdf file ]
Assignment #4 [ pdf file ] ... with answers [ pdf file]
Linear functionals and duality
Assignment #5 [ pdf file ] ... with answers [ pdf file]
Midterm [ pdf file ] ... with answers [ pdf file]
Solution of linear systems of equations
Homogeneous and inhomogeneous systems
Determinants: definition and properties; Permutations and
evaluation
Elementary matrix operations
Invertible matrices; Cramer's rule
Assignment #6 [ pdf file ] ... with answers [ pdf file]
Inner Product Spaces
Inner product
Orthonormal Bases; GramSchmidt; direct decompositions
Bessel, Parseval and CauchySchwartz (in)equalities
Assignment #7 [ pdf file ] ... with answers [ pdf file]
Adjoints
Positive, unitary and normal operators
The spectral theorem
Note:
On the the orthogonal decomposition of Hilbert spaces
[ pdf file ]
Assignment #8 [ pdf file ] ... with answers [ pdf file]
Invariant decompositions
Direct sum decomposition's
Invariant subspaces
Eigenvectors and eigenvalues
Assignment #9 [ pdf file ] ... with answers [ pdf file]
Polynomials: the algorithm of Euclid; Factorization and (multiple) roots
Characteristic polynomial
Triangulation of matrices
Diagonalizable operators
Jordan canonical forms
Assignment #10 [ pdf file ] ... with answers [ pdf file]
CayleyHamilton: characteristic and minimal polynomials; companion matrix
Applications
Note:
Normal matrices are unitarily diagonalizable
[ pdf file ]
Assignment #11 [ pdf file ] ... with answers [ pdf file]
Bilinear forms
General forms
Symmetric forms
Sylvester theorem
Principle coordinates
Note:
Bilinear forms, quadratic forms and congruence
[ pdf file ]
Assignment #12 [ pdf file ] ... with answers [ pdf file]
Final ... [ pdf file ] ... with answers [ pdf file]
Euclidean geometry
3D vectors; scalar and vector products
Lines and relation between lines
Planes: definition and characterizations
Planes and lines
Change of coordinates; translation, rotation and reflection
References
(Partial List)
 Linear Algebra, 2nd Edition, by Hoffman & Kunze, Published by
Prentice Hall. ISBN: 0135367972
 Linear Algebra, 3rd Edition, by S. Lang, Published by SpringerVerlag.
ISBN: 0387964126
 Linear Algebra in Action
by Harry Dym, Graduate Studies in Math v. 78, American Math Society,
2007
 Linear Algebra,
by Peter Lax
 Applied Linear Algebra, by
Gilbert Strang
 Lecture Notes
Eitan Tadmor


