Linear Algebra

Linear algebra is the most applicable area of mathematics. It is one of the fields, that is accepted universally to be the prerequisite to be the in-depth understanding of the machine learning. This field is considered to be the mathematics of data and is especially used in the field of statistics, and used as a tool in Fourier series, computer graphics and so on.

Introduction to Linear Algebra

Linear algebra is the study of linear combinations. It is the study of vector spaces, lines and planes, and some mappings that are required to perform the linear transformations. It includes vectors, matrices and linear functions. It is the study of linear sets of equations and its transformation properties.

Linear Algebra Equations

The general linear equation is represented as

a₁x₁ + a₂x₂……….+a_nx_n = b

Here,

a’s – represents the coefficients

x’s – represents the unknowns

b – represents the constant

There exists a system of linear algebraic equations, which is the set of equations. The system of equations can be solved using the matrices.

It obeys the linear function such as

(x₁,……..x_n) → a₁x₁ +……….+a_nx_n

Vectors

Vectors are things you can add and scalar multiply. Some of the examples of the kinds of vectors addition are as follows :

Numbers: Consider 3 and 5 are numbers. So add it. 3 + 5
n – vectors : 3 – vectors, \(\begin{pmatrix} 1\\ 1\\ 0\end{pmatrix}+\begin{pmatrix} 0\\ 1\\ 1\end{pmatrix}=\begin{pmatrix} 1\\ 2\\ 1\end{pmatrix}\)
Polynomials : If p(x) = 1+x-2x² +3x³ and q(x)= x+3x²-3x³+x⁴, then their sum is p(x)+q(x)= 1+2x+x²+x⁴ is the new polynomial.
Power series : If f(x) = 1+x+(1/2!)x²+(1/ 3!)x³+… and g(x)= 1-x+(1/2!)x²-(1/ 3!)x³+…, then the vector addition is f(x)+g(x)=1+(1/ 2!)x² +(1/ 4!)x⁴+…….is also a power series.
Functions with certain domain: If f(x)= e^x and g(x)= e^-x , then their sum is f(x)+g(x) is a new function

Vectors of different kinds cannot be added. Only the two things of the same kind can be added.

Consider an example with vector addition of two kinds which does not produce any meaning.

\(\begin{pmatrix}{4}\\{5}\end{pmatrix}+e^{x}\)

Linear Function

Linear functions are the functions in which any input say ‘x ‘ is given, the output will be a function of x. That is f(x). Some of the examples of the kinds of vectors that can be rephrased in terms of the function of vectors.

10x = 3, what number x satisfies?
What 3- vector u satisfies, \(\begin{pmatrix} 1\\ 1\\ 0\end{pmatrix}\times u=\begin{pmatrix} 0\\ 1\\ 1\end{pmatrix}\)?
What polynomial p satisfies,\(\int_{-1}^{1}p(y)dy=0\) and \(\int_{-1}^{1}yp(y)dy=1\) ?
What power series f(x) satisfies \(x\frac{d}{dx}f(x)-2f(x)=0\) ?
What number x satisfies 4x² = 1 ?

Here all of the examples given above shows that what vector X satisfies f(X) = B?

Linear Algebra Matrix

Matrices are linear functions of a certain kind. Matrix is the result of organizing information related to certain linear functions. Matrix almost appears in linear algebra because it is the central information of linear algebra. Consider an example,

Question:

A room contains x bags and y boxes of fruits and each bag contain 2 apples and 4 bananas and each box contains 6 apples and 8 bananas. There are a total of 20 apples and 28 bananas in the room. Find the value of x and y.

Solution :

Write the simultaneous equation for the given information that the above condition becomes true.

2x + 6y = 20

4x + 8y = 28

Here the example given above shows the system of linear equations.

Now, write the above equation as an equality between 2- vectors and using the rules, we get

\(\begin{pmatrix} 2x+6y\\ 4x+8y \end{pmatrix}= \begin{pmatrix} 20\\ 28 \end{pmatrix}\) \(x\begin{pmatrix} 2\\ 4 \end{pmatrix}+y\begin{pmatrix} 6\\ 8 \end{pmatrix}=\begin{pmatrix} 20\\ 28 \end{pmatrix}\)

We denote the functions as an array of numbers is called a matrix.

Therefore, the function \(\begin{pmatrix} 2 &6 \\ 4& 8 \end{pmatrix}\) is defined by

\(\begin{pmatrix} 2 &6 \\ 4& 8 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}=x\begin{pmatrix} 2\\ 4 \end{pmatrix}+y\begin{pmatrix} 6\\ 8 \end{pmatrix}\)

Linear Algebra Applications

Here are some of the linear algebra applications as follows

Ranking in Search Engines – One of the most important use linear algebra is in the creation of google. The most complicated ranking algorithm is created with the help of linear algebra.
Signal Analysis – It is massively used in encoding, analyzing and manipulating the signals that can be either audio, video or images etc.
Linear Programming – Optimization is an important application of linear algebra which is widely used in the field of linear programming.
Error-Correcting Codes – It is used in coding theory. If an encoded data is tampered with a little bit and with the help of linear algebra it should be recovered. One such important error-correcting code is called hamming code
Prediction – Predictions of some objects should be found using linear models which are developed using linear algebra.
Facial Recognition- An automated facial recognition technology that uses linear algebraic expression is called principal component analysis.
Graphics- An important part of graphics is projecting a 3-dimensional scene on a 2-dimensional screen which is handled only by linear maps which are explained by linear algebra.

Linear Algebra Problems

Here is a simple example in linear algebra having a system of linear equations.

Question:

Find the value of x, y and z for the given system of linear equations.

2x + y – z = 8

-3x – y + 2z = -11

-2x + y + 2z = -3

Solution:

Given,

2x + y – z = 8

-3x – y + 2z = -11

-2x + y + 2z = -3

The matrix is of the form,

AX = B,

Here , \(A= \begin{bmatrix} 2 & 1& -1\\ -3 & -1& 2\\ -2 & 1 &2 \end{bmatrix}\) \(x=\begin{bmatrix} x\\ y\\ z \end{bmatrix}\) \(B=\begin{bmatrix} 8\\ -11\\ -3 \end{bmatrix}\)

After performing elementary row operation and augmented matrix, it is reduced to the form

A \(\begin{bmatrix} 2 & 1 & -1 & 8\\ -3 & -1 & 2 & -11\\ -2 & 1 & 2 & -3 \end{bmatrix}\)

Now the reduced echelon form of the above matrix is,

A \(\begin{bmatrix} 1& 0 & 0& 2\\ 0 & 1 & 0 & 3\\ 0 & 0 & 1 & -1 \end{bmatrix}\)

Therefore, the unique solution for this is,

X = 2

Y = 3

Z = -1