Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, linear transformations, and systems of linear equations.

This discipline offers a set of tools to describe and manipulate geometric entities and functions algebraically, often through the use of matrices.

It is applied in various fields, including engineering, physics, economics, and statistics. In computer science, it is essential for developing computer graphics algorithms, data analysis, and machine learning.

Here are some key concepts in linear algebra:

• Vectors
  Vectors are mathematical entities with both magnitude and direction. They can be represented as arrays of numbers, which define how to move from one point to another in space.

  Example. An example of a vector in three-dimensional space is: $$ \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix} $$ This vector has three components, representing coordinates along the x, y, and z axes. Here, the vector v points to a location in space 3 units along the x-axis, -2 units along the y-axis, and 5 units along the z-axis, starting from the origin (0, 0, 0). This basic example shows how vectors are used to represent positions or directions in space.

• Vector Spaces
  Vector spaces are collections of vectors (or other objects) that can be added together and multiplied by scalars. A vector space must satisfy certain properties, such as closure under vector addition and scalar multiplication.

  A classic example of a vector space is R², the space of two-component vectors with real values. This is the space of two-dimensional vectors, which can be represented as points or arrows in the Cartesian plane. Each vector in R² has two components, corresponding to the x and y coordinates. $$ \mathbb{R}^2 = \{ (x, y) \mid x, y \in \mathbb{R} \} $$ In this notation, R² is defined as the set of all ordered pairs (x,y), where x and y are real numbers. This vector space includes important properties like closure under vector addition and scalar multiplication, fundamental requirements for any vector space.

• Linear Transformations
  Linear transformations, also known as linear mappings, are a fundamental concept in linear algebra. In simple terms, a linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication.

  Example. Linear transformations include rotations, reflections, dilations, and other geometric transformations that preserve the origin and linear relationships between points.

• Matrices
  Matrices are square or rectangular arrays of numbers representing linear transformations. Operations with matrices, such as addition, multiplication, transposition, determinant calculation, and inversion, are central in linear algebra. The determinant of a matrix provides important information about its properties, such as invertibility. Eigenvalues and eigenvectors of a matrix are crucial in studying its linear transformations.

  Example. Here is an example of a 3×3 matrix: $$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} $$ This matrix consists of three rows and three columns, with the numbers from 1 to 9 arranged in ascending order.

• Systems of Linear Equations
  Systems of linear equations are collections of linear equations that share the same variables. Linear algebra provides tools to analyze and solve these systems.

  Example. Here is an example of a system of linear equations: $$
  \begin{cases}
  x + 2y - 3z &= 7 \\
  3x - y + 5z &= -1 \\
  4x + y + z &= 3
  \end{cases}
  $$ This system comprises three linear equations with three unknowns (x, y, z). The goal is to find the values of x, y, and z that satisfy all three equations simultaneously. This type of system can be solved using methods like Cramer's rule, row reduction, or matrix methods.