Linear Algebra-

Before learning concepts of linear algebra, it is essential to understand what it is besides being a branch of mathematics. This is because this subject accompanies various topics that apply in various fields, including machine learning. Getting the basics of linear algebra also plays a vital role in introducing students to its advanced levels while building their knowledge in this field of study. For this reason, ensure you understand linear algebra definition, basics, and its applications.

Studying linear algebra can be quite challenging for students who fail to comprehend what it entails and the basics that make up the subject. Thus, it is essential to get the right materials to familiarize yourself with linear algebra though getting the best book for linear algebra can also become tricky. That said, here is a manual that helps beginners to learn linear algebra seamlessly.

Linear Algebra

Typically, linear algebra is a mathematics field focusing on linear equations and is often represented in vector spaces and matrices. This is often the general definition of what is linear algebra, which is often central to various math areas. Some of them include modern presentations of geometry, functional analysis, and science and engineering. Besides, linear algebra tends to be the prerequisite to a deeper understanding of machine learning.

Also referred to as the mathematics of data, linear algebra involves linear combinations of data. When looking to create linear algebra equations, you need to create new columns and arrays of numbers. However, you need first to know the concept of matrices and vectors, which are the building blocks of linear algebra. Here at, we’ll provide a complete guide to help you create and solve different linear algebra equations and excel in the subject.

History of Linear Algebra

Rene Descartes began the linear equations systems back in 1637 through geometry coordinates, which has lines and planes representing linear algebra. The first systematic approach to solving linear equations began in 1693 by Leibniz. In 1750, Gabriel Cramer used similar methods to provide explicit solutions to linear systems; today referred to as Cramer’s rule. In 1844, Hermann Grassmann introduced new topics within the subject, while James Joseph Sylvester introduced the term matrix.

Over the years, linear algebra grew with Arthur Cayley introducing matrix multiplication and inverse matrix in 1856 while Peano introduced the first modern and accurate definition of vector space. The linear transformation of finite-dimensional vector spaces emerged by 1900, but linear algebra took its modern approach in the first half of the 20th century. Most previous introductions of the subject were generalized as abstract linear algebra, making these equations more precise and suitable for computer algorithms.

Vector Spaces

Until the 19th century, matrices and systems of linear equations ruled the subject, but modern mathematics presents these equations as vector spaces. This is because it is more synthetic, general, and theoretically simpler but more abstract. Vector spaces usually accompany two elements; vectors represented by V and scalars represented by P. The first operation of vector addition takes two vectors v and u, which creates the third vector v + w. In contrast, second scalar multiplication takes any scalar a any vector and creates a new vector av.

An example of a vector and scalar element is, if u=6, v=11 and u.v=1, find u+v, which signifies a given axiom. Notably, axioms satisfy various addition and scalar multiplication for both arbitrary elements and arbitrary scalars. Remember that specific vector spaces may come in different natures, such as a sequence, function, matrix, or polynomial. Other forms of vector spaces are;

  • Linear Maps: These are mappings between vector spaces commonly used to preserve the vector-space structure. Provided you have two vector spaces V and W over a field F; a linear map remains a map. This map is compatible with additions and scalar multiplication for any vectors of u, v in V, and scalar a in F.
  • Subspaces, Span, and Basis: The study of subsets, referred to as linear subspaces, of vector spaces under induced operations, is vital in different mathematical structures. A linear subspace V over a field F is a subset W of V; hence u + v and au are in W, for each u, v in W, and each a in F. You can also consider a linear combination of sets when forming subspaces.


Matrix linear algebra allows explicit manipulation of finite-dimensional vector spaces and linear maps, bringing a crucial part of linear algebra. For instance, let V be a finite-dimensional vector space over a field F, and (v1, v2….vm) be the basis of V where m is the dimension of V. The map in matrices remains a bijection from Fm, which is the set of sequences of m elements of F, onto V. The equation is hence represented as;

                (a1,…,am)  a1v1 + … amvm

                          Fm  V

This is often the isomorphism of vector spaces, especially if Fm consists of its standard structure of vector space with vector addition and scalar multiplication completed procedurally. Here, the isomorphism allows the vector to be an inverse image represented by either a coordinated vector or a column matrix. Such an approach yields two matrices with corresponding linear maps, and if they are similar, an elementary linear algebra can help prove that one can transform the other.

Linear Systems

A linear system or a system of linear equations is a finite set of linear equations in a calculable set of variables, for example, x1, x2…xn or x, y…z. Linear systems are vital elements in linear algebra as they were used both historically and in the modern linear equations to provide solutions to various problems. An example of a linear system may involve three sets of question such as;

                          2x + y – z = 8

                        -3x – y + 2z = -11

                        -2x + y + 2z = -3

These equations can be arranged in matrices to form three columns to create the right vector. You can also create a homogenous system with the right-hand sides be equated to zero to create a solution using the kernel approach. When two fundamental questions about a linear system involve existence and uniqueness, it follows the same matrix interpretation and applies it to solve the problem.

Endomorphisms and Square Matrices

Linear endomorphism is a linear map mapping a vector space V to itself if it has a basis of m element, often represented by a square matrix of size n. Regarding general linear maps, both square matrices and linear endomorphisms have unique properties, making them suitable for linear algebra and geometric transformations, and quadratic forms.

  • Determinant: The determinant of a rectilinear matrix A involves Sn, which is the group of all variations of n elements. Cramer’s rule applies in such instances when calculating solutions, but when n = 2 or 3, the Gaussian elimination is more useful.
  • Eigenvalues and Eigenvectors: If you have a linear endomorphism of f, a vector of V over a field F, the eigenvector of f becomes a nonzero vector v of V leading to f(v) = av for any scalar a in f. Here, the scalar a is an eigenvalue of f.


A linear form is a linear map of a vector V over a field F to the scalars F field, considered a vector space over itself. It coupes a pointwise addition and multiplication via a scalar called dual space of V, often denoted as V*. Here, the dual map defines two vector spaces over the same field as an induced map usually generalized by adjoint functors. Vector spaces also include additional structures referred to as inner-product spaces, more so for applied linear algebra.

Relationship with Geometry

Linear algebra and geometry have a robust relationship that dates back to 1637, during its introduction of Cartesian coordinates. Now referred to as Cartesian geometry, geometry has points represented by Cartesian coordinates, which is a three-dimensional space. The basic lines and planes in geometry are represented by linear equations when solving these various points.

Again, geometric transformations like reflections, projections, and translations also transform into specified lines and studies based on linear maps. Geometry spaces were defined by axioms associated with lines and planes until the end of the 19th century. Today various academic materials such as Khan academy linear algebra consider geometry a subfield of linear algebra at the elementary level.

Usage and Applications

Learning linear algebra and its applications is vital as it applies in different categories within mathematics. With linear algebra being relevant in nearly all scientific domains, here are some of its application areas;

  • The geometry of ambient space
  • Functional analysis
  • Study of complex systems
  • Scientific computation

Extensions and generalization

  • Module theory: With the existence of multiplicative inverses, you may replace the field of scalars with a ring R giving you a structure referred to as module R or R-.
  • Multilinear algebra and tensors: In multilinear algebra, you can use multivariable linear transformation to map each number separately, creating a dual space that alters the application of linear algebra.
  • Topological vector spaces: Non-finite dimensions allow for additional structure, which creates a normed vector space, inducing a metric and then a topology that defines continuous maps.
  • Homological algebra


Is Linear Algebra Difficult?

Determining how difficult is linear algebra is straightforward because abstract linear algebra is hard, but the mechanics are quite manageable. Using various linear algebra examples and educative materials can help you learn it effectively.

Is Linear Algebra After Calculus?

It is essential to learn calculus first because linear algebra caries more theory and complex concepts, which are abstracts gained during college-level linear algebra.

What Is Linear Algebra Used for?

Linear algebra applies in different domains, including matrices for engineering, graphs and networks, linear algebra for statistics and probability, and computer graphics.

Why Is It Called Linear Algebra?

This is because it is studying equations in straights lines with functions generated from different line points.

Do You Need Calculus for Linear Algebra?

No, linear algebra is quite different from calculus; hence you can readily study Slader linear algebra without considering calculus topics.

Bottom Line

Linear algebra can become quite interesting when you understand what it entails and how to solve assignments and apply the knowledge gained. Here at, we remain dedicated to helping students as you thrive in the subject. If you experience any problems learning linear algebra, contact us to receive professional help today!