Linear Algebra

Linear algebra is the most applicable area of mathematics. Linear algebra is the study of vectors, matrices and linear functions. Linear algebra is the study of linear sets of equations and its transformation properties. The general linear equation is represented as

a1x1 +……….+anxn = b

It obeys the linear function such as

(x1,……..xn ) \(\rightarrow\) a1x1 +……….+anxn

Linear Algebra Basics

The Basics of linear algebra are as follows

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)= ex and g(x)= e-x , then their sum is f(x)+g(x) is a new function

Vectors of different kinds can not 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 Functions

In linear algebra, linear functions are the functions in which any input say ‘x ‘ is given, an 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 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 ?

Matrix

Matrices are linear functions of certain kind. Matrix are the result of organizing information related to certain linear functions. Matrix almost appear 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 contains 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 and its 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 is developed using linear algebra.
  • Facial Recognition- An automated facial recognition technique 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 is explained by linear algebra.

Linear Algebra Problems

Here is a sample example in linear algebra having 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

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Practise This Question

The general solution of dydxyx=y2x2 is