Linear Algebra: Linear Functions, Vectors, and Matrices

Linear algebra is a branch of mathematics that deals with the study of vectors and linear functions.

• It is a key concept used in almost every mathematical discipline.
• It is considered a basic concept in the modern presentation of geometry.
• It is mostly used in Physics and Engineering to define basic objects such as planes, lines, and rotations of the object.
• Linear algebra allows us to model a wide range of natural phenomena with remarkable computational efficiency.

Key Terms: Algebra, Linear Algebra, Linear algebra equations, Vector Spaces, Linear Functions, Linear Algebra Matrix, Matrices, Vector axioms

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What is Linear Algebra?

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Linear algebra is a branch of mathematics, that studies linear functions within vector spaces.

• Organized information about these functions forms matrices.
• It includes vector spaces, vectors, linear functions, systems of linear equations, and matrices.
• These concepts are essential for related topics like geometry and functional analysis.

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Linear Algebra Equation

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The general linear equation is represented by

Where

• a₁, a₂, ……., a_n are the coefficients
• _x1, x₂, ……., x_n are unknowns
• b is a constant.

Linear algebra deals with systems of linear algebraic equations, which are sets of equations involving linear relationships between variables. These systems can be efficiently solved using matrices.

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Linear Algebra Topics

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The topics covered under linear equations are as follows:

• Eigenvalues and eigenvectors
• Euclidean vector spaces
• Linear dependence and independence
• Linear transformations
• Singular value decomposition
• Projections
• Orthogonal matrices
• Solving systems of equations with matrices
• Matrix inverses and determinants
• Mathematical operations with matrices (i.e. addition, multiplication)
• Positive-definite matrices

All these topics mentioned above in linear algebra are classified into three broad categories. These are

• Vector spaces
• Linear Functions
• Matrix

All three concepts are connected in such a way that they can be used mathematically to construct a system of linear equations. In general, vectors are elements that can be added, and linear functions are the functions of vectors that include vector addition.

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Vector Spaces

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Linear algebra studies vector spaces and the transformations (changes) that occur between them using linear transformations.

A vector is a mathematical object with both magnitude (size) and direction.

• Scalars are generally real numbers.
• In some cases, they can be non-real numbers, i.e. complex numbers, or naturally any field.
• Vector addition and scalar multiplication must follow specific rules called vector axioms.
• The terms "real vector space" and "complex vector space" are used to define scalars are real numbers or complex numbers, respectively.

Let V be a vector space with the elements a, b, c, and scalars m, n over any field F, then the vector axioms are:

• Commutative Addition: a + b = b + a
• Associativity Addition: a + (b + c) = (a + b) + c
• Additive Identity: a + 0 = 0 + a = a, where 0 is an element in V called zero vector.
• Additive Inverse: a + (-a) + (-a) + a = 0, a, -a belongs to V.

The above four axioms define the vector spaces as the abelian group.

• Other axioms include the distributivity of scalar multiplication in correspondence to the vector addition and so on.
• For example, m(a) = ma; n(a + b) = a + n.

Elements within a specific vector space can have diverse characteristics.

• For example, they can be sequences, functions, polynomials, or even matrices.
• Linear algebra focuses on the properties these elements share, which are common to all vector spaces.
• A linear map can be written for two vector spaces, V and W, over a field F.
• This is also known as linear transformation or mapping of vector spaces.

Thus, it is given by:

T: V → W

This lets us to write the addition and scalar multiplication of elements, such as:

T(a + b) = T(a) + T(b)

T(ma) = mT(a)

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Linear Functions

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Linear algebra uses vectors to form linear functions. Some examples of vectors can be rephrased or represented using these linear functions of vectors.

Mathematically, a linear function is defined as:

A function L: Rⁿ → R^m is linear if

1. L(x + y) = L(x) + L(y)
2. L(αx) = αL(x)

for all x, y ∈ Rⁿ, α ∈ R

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Linear Algebra Matrix

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Matrices play an important role in linear algebra. They can be considered as a special type of linear function, with information related to these functions arranged into a matrix.

Z is an m × n matrix, then we get a linear function L: Rⁿ → R^m by defining

L(x) = Zx

or 

Zx = B

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Numerical Linear Algebra

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Numerical linear algebra, sometimes referred to as applied linear algebra, focuses on using matrix operations to design efficient and accurate computer algorithms for solving problems arising in continuous mathematics. 

• Many matrix decomposition methods are used in numerical linear algebra to find solutions to usual linear algebraic problems.

Such as least-squares optimization, Eigenvalue location, and system of linear equations solving.

• Some of the matrix decomposition methods used in numerical linear algebra are

Eigen decomposition, single value decomposition, and QR factorization.

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Applications of Linear Algebra

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The are many applications of Linear algebra in real life, some of them are given below.

• Graphics: It can be used to project the 3D scene on any 2D screen using different linear maps as explained in linear algebra.
• Prediction: Predictions can also be made by making linear algebraic models.
• Facial Recognition: Automated Facial Recognition can also be done using linear algebra, referred to as principal component analysis.
• Signal Analysis: Audio, video, or images all types of signals can be encoded, analyzed, or manipulated using linear algebra.
• Linear Programming: In the field of Linear Programming, optimization is a very crucial step wherein Linear algebra is again a great help.
• Ranking in Search Engines: Linear Algebra is also used in the creation of Google ranking algorithms.
• Error-Correcting Codes: Linear algebra also helps in recovering the tampered codes.

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Solved Examples

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Things to Remember

• Linear algebra is referred to as the study of linear combinations.
• It studies the vector spaces, lines and planes, and some mappings that are required to perform the linear transformations.
• Linear algebra is classified into three broad categories: Vector spaces, Linear Functions, and Matrix.
• A general linear equation is given by a₁x₁ + a₂x₂……….+ a_nx_n = b.
• A vector space can be defined as a collection of objects called vectors.
• Linear algebra uses vectors to form linear functions.
• Matrixes are a systematic and efficient method of representing the basic information of linear algebra.