PROJECT REPORT

ON

Bases And Dimensions

For the Patial Fulfillment of Degree of B.sc Mathematics

M.B.GOVT.P.G COLLEGE, HALDWANI (NAINITAL)

PROJECT GUIDE: SUBMITTED BY:

Dr. Rakesh Kumar NAME – Rohit Singh

Course – B.sc

Roll No. - 220130270357

 Department of Mathematics

CERTIFICATE

It gives us great pleasure to testify the ability of Rohit Singh S/O

Mr. Keshar Singh, who is presently a student of B.Sc. fifth semester in
the Department of Mathematics, M.B. Govt.P.G.College, Haldwani
(Nainital).

Rohit Singh has successfully completed the project work on

“Bases And Dimensions” to fulfill the partial requirement of Master’s
degree in mathematics. She is a consistently hard working, honest,
sincere and conscientious person.

Name of H.O.D Name of supervisor

Dr. D.K Tiwari Dr. Rakesh Kumar

 ACKONWLEDGEMENT

I wish to thank my parents for tremendous

contribution and support both morally and financially
towards the completion of this project.

I am also grateful to my project supervisor Dr. Rakesh

Kumar who without her help and guidance this project
would not have been completed.

I also show my gratitude to my friends and all who

contributed in one way or the other in the course of the
project.

CONTENTS
1. Discovery of basis and
dimensions
2. Introduction
3. Introduction to basis and
dimensions
4. Basis
5. Properties of basis
6. Finding of basis
7. Applications of basis
8. Change of basis
9. Visual representations
10. Conclusion of basis
11. Refrences of basis
12. Dimensions
13. Historical background
14. Types of dimensions
15. Applications
16. Visualization techniques
17. Philosophical
consideration
18. Conclusion of dimensions
19. Refrences of dimensions
20. Additional tips
21. Summary
22. References

DISCOVERY OF BASES
AND DIMENSIONS .
The concepts of bases and
dimensions in the context of
vector spaces are fundamental
ideas in linear algebra, and their
development involved
contributions from multiple
mathematicians over time.

Bases:
The concept of a basis for a
vector space was developed as
part of the broader study of
vector spaces in linear algebra.
While the formalization of the
notion of a basis can be
attributed to the work of
Giuseppe Peano (1890s) and
Ernst Eduard Kummer (mid-
1800s), the modern definition of a
basis, as a set of vectors that are
linearly independent and span
the vector space, became clear
with the work of Arthur Cayley
and James Joseph Sylvester in the
19th century.
The systematic study of vector
spaces, including the notion of a
basis, was further developed by
David Hilbert in the early 20th
century. The idea of a basis has
become central to the structure
of vector spaces in modern linear
algebra.

Dimension:
The idea of dimension (the
number of vectors in a basis)
came naturally with the
development of the concept of a
basis. The term “dimension” as
we use it today was formalized in
the late 19th and early 20th
centuries as the number of
elements in a basis for a vector
space. This is closely related to
the work of Giuseppe Peano,
David Hilbert, and others who
formalized the properties of
vector spaces and their
dimensions in the context of
functional.

The discovery and formalization

of bases and dimensions were
gradual processes involving
contributions from
mathematicians such as Peano,
Cayley, Sylvester, and Hilbert in
the 19th and early 20th centuries.

These concepts are foundational

to linear algebra and are still
central to modern mathematics,
especially in fields like functional
analysis, geometry, and computer
science.

INTRODUCTION :
. Importance of basis and
dimension
. Objectives of the research

INTRODUCTION TO
BASIS AND DIMENSION :
BASIS :
A basis of a vector space is a set
of vectors that are linearly
independent and span the space.
Every vector in the space can be
expressed as a linear combination
of the basis vectors.

A basis of a vector space is a set

of vectors that satisfies two key
properties:

1. Linear Independence : The

vectors in the basis must be
linearly independent,
meaning no vector in the set
can be expressed as a linear
 combination of the others.
Formally, for a set of vectors .

Formula:-

i.e. c1v1 + c2v2 ……… cnvn

=0

c1 = c2 …………..= cn = 0

2. Spanning Set: The vectors

must span the vector space,
meaning any vector in the
space can be expressed as a
linear combination of the
basis vectors.

Formula:-
 V = a1v1 + a2v2 ………. anvn

For some scalers a1 , a2 ,

…….. an (V1,V2………Vn)
Are basis of Space V .

Properties of a Basis :

•Uniqueness of Representation:
Each vector in the vector space
can be uniquely expressed as a
linear combination of the basis
vectors. This means that the
coefficients in the linear
combination are unique.

•Dimension:
The number of vectors in a basis
defines the dimension of the
vector space. All bases of a vector
space have the same number of
elements.

•Linear Independence:
A basis is a set of vectors that are
linearly independent, meaning no
vector in the set can be
expressed as a linear combination
of the others.
• Spanning:
A basis spans the vector space,
meaning every vector in the
space can be represented as a
linear combination of the basis
vectors.

•Change of Basis:
Any basis can be transformed into
another basis through a linear
transformation.

•Existence:
Every finite-dimensional vector
space has at least one basis.
Infinite-dimensional spaces may
also have bases, but their
construction can be more
complex.
•Subset Property:
Any linearly independent subset
of a vector space can be extended
to a basis of that space.

These properties are fundamental

in understanding the structure
and behavior of vector spaces in
linear algebra.

Uniqueness of Representation:
Each vector in the vector space
can be uniquely expressed as a
linear combination of the basis
vectors. This means that the
coefficients in the linear
combination are unique.
Dimension: The number of
vectors in a basis defines the
dimension of the vector space. All
bases of a vector space have the
same number of elements.

Linear Independence: A basis is a

set of vectors that are linearly
independent, meaning no vector
in the set can be expressed as a
linear combination of the others.

Spanning: A basis spans the

vector space, meaning every
vector in the space can be
represented as a linear
combination of the basis vectors.
Change of Basis: Any basis can be
transformed into another basis
through a linear transformation.
This is useful in applications like
computer graphics and machine
learning.

Existence: Every finite-

dimensional vector space has at
least one basis.
Infinitedimensional spaces may
also have bases, but their
construction can be more
complex.

Subset Property: Any linearly

independent subset of a vector
space can be extended to a basis
of that space.
These properties are fundamental
in understanding the structure
and behavior of vector spaces in
linear algebra.

Finding a Basis :
Present methods for finding a
basis for a given vector space:
Using row reduction (Gaussian
elimination).
Identifying pivot columns in a
matrix. The Gram-Schmidt
process for orthonormal bases.

Applications of Basis :
Discuss practical applications of
bases in various fields:
Computer graphics (coordinate
transformations).
Data science (dimensionality
reduction, PCA).
Engineering (signal processing).

Change of basis :
Explain what changing a basis
means and why it’s useful.
Provide an example of how to
change from one basis to another.

Visual Representation :
Include diagrams illustrating
vector spaces and bases.
Graphs showing linear
combinations and spans.
Conclusion :
Summarize the importance of
understanding bases in linear
algebra. Reflect on how this
knowledge can be applied in real-
world situations.

References :
List textbooks, academic papers,
and online resources for further
reading.

DIMENSIONS :
The number of vectors in a basis
is called the dimension of the
vector space. All bases of a vector
space have the same number of
vectors, which is a fundamental
result in linear algebra. The
dimension of a vector space is the
number of vectors in a basis for
that space. It indicates the
minimum number of coordinates
needed to specify any vector in
that space. In the section on
spanning sets and linear
independence, we were trying to
understand what the elements of
a vector space looked like by
studying how they could be
generated. We learned that some
subsets of a vector space could
generate the entire vector space.
Such subsets were called
spanning sets. Other subsets did
not generate the entire space,
but their span was still a
subspace of the underlying vector
space. In some cases, the number
of vectors In such a set was
redundant in the sense that one
or more of the vectors could be
removed ,without changing the
span of the set. In other cases,
there was not a unique way to
generate some vectors in the
space. In this section, we want to
make this process of generating
all the elements of a vector space
more reliable, more client .

EXAMPLE :
Standard Basis in R²: The
standard basis for the 2-
dimensional real space

Historical background :
Early Concepts: Discuss early
ideas of dimensions :-
a. 1D
b. 2D
c. 3D

Notable Mathematicians: Mention

key figures like Euclid, Riemann,
and others who contributed to
the concept of dimensions.

This was the types of an early age

dimensions classification .
So let’s talk about the present
day types of dimensions

Types of dimensions:

Euclidean Dimensions:

1D: Lines, intervals.

2D: Planes, polygons.
3D: Solids, volumes.

Higher Dimensions:

4D and Beyond: Introduce

concepts of hypercubes and
applications in theoretical
physics.
Mathematical Representation :
Coordinate Systems: Explain
Cartesian and polar
coordinates.
Vector Spaces: Discuss how
dimensions relate to vector
spaces in linear algebra.
Fractal Dimensions: Introduce the
concept
of fractals and their non-integer
dimensions.

Applications :
Geometry and Topology: How
dimensions influence shapes and
spaces.
Physics: The role of dimensions in
theories like string theory.
Computer Science: Dimensionality
reduction techniques in data
analysis (e.g., PCA).
Visualization Techniques :
Graphs and Charts: Methods to
visualize higher dimensions.
Dimensional Analysis: Tools used
to understand complex
dimensions in practical scenarios.

Philosophical Considerations :
Nature of Reality: Explore
philosophical implications of
multiple dimensions.
Theoretical Physics: Discussion
on the nature of time and space.
Conclusion :
Summary: Recap the significance
of understanding dimensions.
Future Directions: Suggest areas
for further research, such as
quantum dimensions or advanced
geometric theories.

References :
Compile a list of books, articles,
and papers used for research.

Additional Tips:
Include diagrams and visual aids
to enhance understanding.
Consider real-world examples to
illustrate abstract concepts.
Engage with contemporary
research to provide current
perspectives on dimensions.

SUMMARY :

In summary, a basis is a
foundational concept in linear
algebra that provides a
framework for expressing vectors
within a vector space. It ensures
that every vector can be
represented uniquely as a
combination of basis vectors,
which is crucial for both
theoretical exploration and
practical applications across
various fields of mathematics,
science, and engineering. Know
and understand the definition of a
basis for a vector space.Know
what the dimension of a vector
space is know what the
coordinates of a vector relative to
a given basis are. Given a set of
vectors In a vector space, be able
to tell if that set is a basis for the
vector space. Know the standard
basis for common vector spaces
such as Rn, Mn, Pn for every
positive integer n. Be able to and
the basis of subspaces given the
description of a subspace. Be able
to and the coordinates of any
vector relative to a given basis.
Be able to and the dimension of a
vector space. Know and
understand the difference
between a finite-dimensional and
infinite dimensional vector space.
It ensures that every vector can
be represented uniquely as a
combination of basis vectors,
which is crucial for both
theoretical exploration and
practical applications across
various fields of mathematics,
science, and engineering.

REFERENCES

• “Elemente der Algebra” (1888)

by Giuseppe Peano .
• “A Treatise on the Theory of
Determinants” (1848) by James
Joseph Sylvester .
• “Mathematical Contributions to
the Theory of Evolution” (1867)
by Arthur Cayley .
• “Foundations of Geometry”
(1899) by David Hilbert .
• “Linear Algebra” by Kurt
Magnus .
• “Abstract Algebra” by David S.
Dummit and Richard M. Foote .