What are you going to learn?
Content
Model and systematically solve systems of linear equations using matrix notation. Demonstrate factual knowledge of the fundamental concepts of spanning, linear independence, and linear transformations. Use matrix algebra to analyze and solve equations arising in many applications that require a background in linear algebra. Utilize vector space terminology and describe how closely other vector spaces resemble Rn. Dissect the action of a linear transformation into elements that are easily visualized using the basic concepts of eigenvectors and eigenvalues.
Chapter 1. Vectors
Definition
Scalar Product
Norm of a Vectors
Parametric Lines
Planes
Definition
Operations with Matrices
Row Operations
The Inverse Matrix
Chapter 2. Matrices and Linear Equations
Chapter 3. Vector Spaces
Definition. Subspaces
Linear Combination, Linear Independence
Basis, Dimension
Finding the Basis of a Linear Space
Coordinates. Changes of Basis
Chapter 4. Linear Mappings
Definition. Linear Mappings
Kernel and Range of a Linear Transformation
Range and Linear Equations
Matrix Associated with a Linear Transformation
Change of Basis
Chapter 5. Composition and Inverse Mappings
Graphs Terminology
Digraphs and Connectivity Problems
Chapter 6. Scalar Products and Ortogonality
Inner Product. Length. Orthogonal Vectors. Triangle Inequality
Cauchy-Schwartz Inequality
Orthonormal Basis, Gram-Schmidt Process
Chapter 7. Determinants
Determinant of 2x2, 3x3 Matrices
Minors and Cofactors of a Square Matrix
Cofactor Theorem
Properties of Determinants
Cramer's Rule
Chapter 8. Eigenvectors and Eigenvalues
Definition
Diagonalization
Symmetric Matrices and Orthogonal Diagonalization
Bibliography
Lang, S. Introduction to Linear Algebra. 2th ed, Springer, 2000.
Larson, R. Elementary Linear Algebra. 8th ed, Cengage Learning, 2017.
