**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.

. 2th ed, Springer, 2000.*Introduction to Linear Algebra*Larson, R.

*Elementary Linear Algebra**. 8th ed, Cengage Learning, 2017*.