Linear Equations and Linear Transformations

- Overview

Linear algebra is a branch of mathematics that focuses on the study of vectors, vector spaces, and linear transformations. It deals with linear equations, linear functions, and their representation through matrices and determinants. It has a wide range of applications in the fields of physics and mathematics. It is a fundamental concept in machine learning and data science.

Linear equations and linear transformations are fundamental concepts in linear algebra. A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable raised to the power of one. 

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. These concepts are closely related, as linear transformations can be represented by matrices, which in turn can be used to solve systems of linear equations. 

In essence: Linear equations form the basis for understanding linear transformations, and linear transformations provide a powerful tool for analyzing and solving systems of linear equations, as well as for understanding geometric transformations in vector spaces.

Please refer to the following for more information:

• Wikipedia: Linear Equation
• Wikipedia: Linear Transformation

- Linear Equations

A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form Ax + B = 0. Here, x is a variable, A is a coefficient and B is constant.

Definition:
A linear equation is an equation that can be written in the form a₁x₁ + a₂x₂ + ... + a_ix_i = b, where a_i and b are constants, and x_i are variables. 

Examples:
2x + 3y = 5, y = 2x + 1, x - y + z = 0 are all examples of linear equations.

Graphical Representation: 

In two dimensions, linear equations represent straight lines. In three dimensions, they represent planes. 

Systems of Linear Equations:
A set of two or more linear equations with the same variables is called a system of linear equations.

- Linear Transformations

In linear algebra, a linear transformation (or linear map) is a function that moves from one vector space to another. It also respects the underlying structure of each vector space. A linear transformation is also known as a linear operator or map. 

A transformation is linear if it satisfies the following two properties: 

• Additive: The output is the same if the numbers are added first and then transformed, or if they are transformed and then the transformations are added together.
• Scalar: For all vectors →x and all scalars k, T(k→x) = kT(→x).

Examples of linear transformations include the zero transformation and the identity transformation. The zero transformation is defined by T(→x) = →(0) for all →x. The identity transformation is defined by T(→x) = →(x). 

Matrices can be used to perform a wide variety of transformations on data.

Definition:
A linear transformation T from a vector space V to a vector space W (denoted as T: V -> W) is a function that satisfies two properties:

• Additivity: T(u + v) = T(u) + T(v) for all vectors u and v in V.
• Homogeneity: T(ku) = kT(u) for all vectors u in V and all scalars k.

Examples:

• Identity Transformation: T(v) = v for all vectors v.
• Zero Transformation: T(v) = 0 for all vectors v.
• Scaling: T(v) = cv, where c is a scalar.
• Rotation: Rotating a vector by a certain angle.
• Reflection: Reflecting a vector across a line or plane.

Representation by Matrices:
A linear transformation from Rⁿ to Rᵐ can be represented by an m x n matrix. If A is the matrix representing T, then T(v) = Av for any vector v in Rⁿ.

Relationship to Systems of Equations:
A system of linear equations can be expressed as a matrix equation Ax = b, where A is the matrix of coefficients, x is the vector of unknowns, and b is the vector of constants. Finding the solution to the system is equivalent to finding the vector x that satisfies the equation T(x) = b, where T is the linear transformation represented by A.