What is Linear Algebra?

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Who Invented Linear Algebra?

A mathematician named Hermann Grassmann published a book called ‘Theory of extension’ in 1844. Today these topics are called linear algebra.

What Exactly is Linear Algebra?

Suppose that you are buying three chocolates for your friend on his birthday. You can either buy the chocolates one by one or buy all three at a time. In both of these cases, you are buying three in all. The difference between both the cases is that in the first case you are buying 3 times one chocolate and in the second case you are buying 3 chocolates at one time. Both of these give the same output. Let the letter ‘x’ represent the number of chocolates and ‘a’ represent the number of times the purchase is made. It can be mathematically expressed as follows. Since this is linear let ‘F’ denote a linear function.

F (ax) = a F(x)

Suppose that you are buying two different items in a shop. If one item costs 50 bucks and the other item costs 10 bucks, how much should you pay the shopkeeper? In total, you have to pay 60 bucks. You can also pay 50 bucks first and 10 bucks later. But the total amount you have to pay is 60 bucks in both cases. Let ‘x’ denote the first item and ‘y’ denote the second item. It is mathematically represented as follows. Here also the function is linear.

F(x + y) = F(x) + F(y)

These linear functions are called linear transformations or linear mapping if it maps one vector space to another vector space but the two vector spaces should be over the same field.

Vector Space

Consider a set whose elements are vectors. This set is called a vector space if it satisfies certain properties. We call an object a vector if it has both magnitude and direction. But scalars have only magnitude.

• When we add two vectors from this set, the result should also be a vector belonging to this same set, and the addition of the vectors can be carried in any order. That is, if ‘x’ and ‘y’ are two vectors, then x + y = y + x. The same condition holds even when three vectors are added. That is,

(x + y) + z = x + (y + z)

• Adding any vector to 0 gives back the same vector. It is called a zero vector. That is,

a + 0 = a

• Adding any vector to its inverse element gives the zero vector. That is

a + (-a) = 0

• Let ‘s’ denote a scalar. If ‘s’ is multiplied by the sum of ‘a’ and ‘b’, the result is the sum of ‘s’ multiplied by ‘a’ and ‘s’ multiplied by ‘b’. That is

s (a + b) = sa + sb

Also, the elements in the vector space can be a matrix or polynomial.

Linear Combinations

Linear algebra in mathematics is a study about linear combinations. Suppose that ‘x’ and ‘y’ are two vectors and ‘a’ and ‘b’ are two scalars. If a vector can be written in the form ax + by, then the vector is said to be written in the linear combination of the vectors. Here, the terms are multiplied by a constant term called scalars and then added.

Matrices and Linear Algebra

Suppose that there are ‘x’ packs and ‘y’ boxes in a room and each pack contains 2 chocolates and 4 biscuits, and each box contains 9 chocolates and 8 biscuits. Suppose that there are 25 chocolates and 28 biscuits in the room. What is the value of ‘x’ and ‘y’.

It is easy to find the value of ‘x’ and ‘y’ if the given information is written as a linear equation.

2x + 9y = 25

4x + 8y = 28

$\left(\begin{array}{c}2\\ 4\end{array}\right)x+\left(\begin{array}{c}9\\ 8\end{array}\right)y=\left(\begin{array}{c}25\\ 28\end{array}\right)$

This can also be written in the matrix form as follows.

$\left(\begin{array}{cc}2& 9\\ 4& 8\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}25\\ 28\end{array}\right)$

Any linear map which is over a finite-dimensional space can be represented in the form of a matrix.

Eigenvalues of Matrices

Eigenvalues are a set of scalar values. Eigenvalues of a matrix are nothing but the values of $\lambda$ which satisfy the equation $\left|A-\lambda I\right|=0$. Here, A is a square matrix, I is an identity matrix, and $\lambda$ is any scalar value. I and A are n by n matrices. Since Eigenvalues are scalar, they tell about the magnitude of the linear map. Eigenvalues can be equal to 0.

Eigenvector of Matrices

Let matrix A be a square matrix with n rows and n columns. Multiply this matrix A by a vector that is not equal to 0. What is the result? It would be a scalar multiple of the vector.

All the vectors, v, corresponding to the eigenvalues $\lambda$ of matrix A which satisfy the equation $Av=\lambda v$ are called the eigenvectors of matrix A.

An eigenvector cannot be equal to 0. An eigenvector gives the direction in which the linear map acts.

Applications of Eigenvectors

The concept of eigenvectors is used in quantum mechanics and other fields in physics.

In mathematics, eigenvectors are used to solve linear equations in differential calculus.

How is Linear Algebra Used in Machine Learning?

When you collect data, you would want to put the values in a table with rows and columns. Linear algebra has great significance here.

Cutting or cropping of images is done by applying linear algebra concepts. Each pixel is like a cell representing the entries of a matrix.

Where to Use Linear Algebra?

• Rotations in 3-dimensional space are very much tougher to calculate than the rotations in 2-dimensional space. Consider a rotation whose center is the origin. Here, the rotations are linear transformations. It is determined by the orthogonal matrices.  The determinant value of orthogonal matrices is always equal to 1 or -1. By saying that the determinant value equals 1, we mean that the direction of the rotation is preserved. Thus, the concept of linear algebra is applied here.
• Linear algebra helps in finding the solution to the differential equations.
• Given three non-collinear points, linear algebra helps in finding the equation of the circle which goes to these points.
• The face recognition technology uses the concept of linear algebra.
• For projecting a 3-D picture on a 2-D screen we need a linear map. This is studied in linear algebra.

Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for      

• B.Sc. Mathematics
• M.Sc. Mathematics