Linear algebra is one of the important branches of mathematics. Linear
algebra is basically the study of vectors and linear functions. It is a key
concept for almost all areas of mathematics. Linear algebra is considered
a basic concept in the modern presentation of geometry. It is mostly used
in Physics and Engineering as it helps to define the basic objects such as
planes, lines and rotations of the object. It allows us to model many natural
phenomena, and also it has a computing efficiency. In this article, you are
going to learn about the basic introduction, its components,
problems, linear equations and its applications.
Table of Contents:
•
•
Introduction
•
Topics
•
Vectors
•
Linear Function
•
Matrix
•
Numerical Linear Algebra
•
Applications
•
Problems
•
Practice problems
•
FAQs
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……….+anxn = 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,……..xn) → a1x1 +……….+anxn
Linear Algebra Topics
The most important topics covered in the linear algebra includes:
•
Euclidean vector spaces
•
Eigenvalues and eigenvectors
•
Orthogonal matrices
•
Linear transformations
•
Projections
•
Solving systems of equations with matrices
•
Mathematical operations with matrices (i.e. addition, multiplication)
•
Matrix inverses and determinants
•
Positive-definite matrices
•
Singular value decomposition
•
Linear dependence and independence
Here, the three main concepts which are the prerequisite to linear algebra
are explained in detail. They are:
•
Vector spaces
•
Linear Functions
•
Matrix
All these three concepts are interrelated such that a system of linear
equations can be represented using these concepts mathematically. In
general terms, vectors are elements that we can add, and linear functions
are the functions of vectors that include the addition of vectors
Vector Spaces
As we know that linear algebra deals with the study of vector spaces and
the linear transformations between them. By the definition of vector, it is a
physical quantity that has both magnitude and direction. A vector space is
defined as the collection of objects called vectors, which may be added
together and multiplied (i.e. scaled) by numbers, called scalars.
Generally, real numbers are taken as scalars, but there exist vector spaces
with scalar multiplication by non-real numbers, i.e. complex numbers, or
naturally any field.
The operations such as vector addition and scalar multiplication must
satisfy specific requirements, called vector axioms. Generally, the terms
real vector space and complex vector space are used to define that the
scalars are real or complex numbers, respectively.
Suppose V be any vector space with elements a, b, c and scalars m, n over
a field F, then the vector axioms are given by:
•
Commutative of addition: a + b = b + a
•
Associativity of 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.
These four axioms define that the vector space V is an abelian group under
addition.
Other axioms include distributivity of scalar multiplication with respect to
vector addition and field addition, identity element of scalar multiplication
and so on.
For example, m(a) = ma; n(a + b) = na + nb
An element of a specific vector space may have different characteristics.
For example, the elements can be a sequence, a function, a polynomial or
a matrix. Linear algebra is affected by those properties of such things that
are common or familiar to all vector spaces.
A linear map can be written for two given vector spaces namely V and W
over a field F. This is sometimes referred to as linear transformation or
mapping of vector spaces. Thus, it is given by:
T: V → W
This allows us to write the addition of scalar multiplication of elements
such as:
T(a + b) = T(a) + T(b)
T(ma) = mT(a)
Linear Function
A linear function is an algebraic equation in which each term is either a
constant or the product of a constant and a single independent variable of
power 1. In linear algebra, vectors are taken while forming linear functions.
Some of the examples of the kinds of vectors that can be rephrased in
terms of the function of vectors.
Mathematically, linear function is defined as:
A function L : Rn → Rm is linear if
(i) L(x + y) = L(x) + L(y)
(ii) L(αx) = αL(x)
for all x, y ∈ Rn, α ∈ R
Example:
Show that the function �:�2→�3 given by �(�)=[�1+4�23�1−�2�2] is linear.
Solution:
For any x, y ∈ R2, we have
For any x ∈ R2 and α ∈ R, we have
Therefore, L is a linear function.
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.
Mathematically, this relation can be defined as follows.
A is an m × n matrix, then we get a linear function L : Rn → Rm by defining
L(x) = Ax
or
Ax = B
Go through the example given below to understand this mapping in detail.
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 equality between 2- vectors and using
the rules, we get
(2�+6�4�+8�)=(2028)
�(24)+�(68)=(2028)
We denote the functions as an array of numbers is called a matrix.
Therefore, the function (2648) is defined by
(2648)(��)=�(24)+�(68)
Numerical Linear Algebra
Numerical linear algebra is also known as the applied linear algebra.
Applied linear algebra deals with the study of how matrix operations can
be used to create computer algorithms, which helps to solve the problems
in continuous mathematics with efficiency and accuracy. In numerical
linear algebra, many matrix decomposition methods are used to find the
solutions for common linear algebraic problems like least-square
optimization, locating Eigenvalues, and solving systems of linear equations.
Some of the matrix decomposition methods in numerical linear algebra
include Eigen decomposition, Single value decomposition, QR factorization
and so on.
Linear Algebra Applications
Here, some of the linear algebra applications are given as:
•
Ranking in Search Engines – One of the most important applications of 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 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.
Also, read: Linear Equation Applications
Linear Algebra Problems
Linear algebra problems include matrices, spaces, vectors, determinants,
and a system of linear equation concepts. Now, let us discuss how to solve
linear algebra problems.
Example 1:
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 ,
�=[21−1−3−12−212]
�=[���]
�=[8−11−3]
After performing elementary row operation and augmented matrix, it is
reduced to the form
�[21−18−3−12−11−212−3]
Now the reduced echelon form of the above matrix is,
�[10020103001−1]
Therefore, the unique solution for this is,
x=2
y=3
z = -1
Example 2:
Find the value of x, given that |3��1|=|3241|
Solution:
Given that:
|3��1|=|3241|
Thus, the above determinant can be equated as:
3 – x2 = 3 – 8
3 – x2 = -5
-x2 = -5-3
-x2 = -8,
Now, multiply both sides by -1, we get
x2 = 8
x = ± 2√2
Therefore, the value of x is ± 2√2.
Also, learn: Solving Linear Equations
Practice Problems on Linear Algebra
1.
Solve the following pair of linear equations using the substitution method: 7x15y=2, x+2y=3.
2. Solve the equations 2x+3y=11, and 2x-4y=-24. Also, determine the value of “m”
for which y=mx+3.
3. Solve the following equations using the substitution method and elimination
methods: 3x + 4y = 10 and 2x – 2y = 2.
For more related articles on the system of linear equations, register with
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Frequently Asked Questions on Linear Algebra
Q1
What is meant by linear algebra?
Linear algebra is the study of lines, planes, vector spaces, mappings, which
are required for the linear transformation.
Q2
Mention the applications of linear algebra?
Some of the real-life applications of linear algebra include facial
recognition, testing of coding in software engineering, ranks in the search
engine, graphics and so on.
Q3
What is called a linear function?
A linear function is a kind of algebraic equation, in which each term is a
constant or a product of constant and a single independent variable of
power 1.
Q4
What is meant by a vector?
A vector is a quantity that has two independent properties called
magnitude and direction.
Q5
What is meant by vector space?
A vector space consists of a set of objects called vectors, which can be
added together and multiplied by the numbers called scalars.
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