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History and applications of vector space and linear transmissions
(ASSIGNMENT # 2 SEMESTER Spring-2020)
Submission Date (April 10, 2020)
BY ROLL#
Mehak Usman
17451509-098
MATH-319 (Linear Algebra-II)
BS-Mathematics (Section-B)
Submitted To
Dr. Shahida Bashir
Department of Mathematics
BS (6th)
UNIVERSITY OF GUJRAT
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CONTENTS
VECTORS
➢ HISTORY OF VECTOR SPACES______________________________3
• Early years____________________________________________3
• Middle years___________________________________________3
• In the 90’s_____________________________________________4
➢ APPLICATIONS OF VECTOR SPACES_________________________4
• Distributions___________________________________________5
• Fourier series___________________________________________5
• Differential geometry____________________________________6
• Quantum mechanics_____________________________________6
• Circuit theory__________________________________________7
• CAD_________________________________________________7
• MATLAB_____________________________________________7
• Least square___________________________________________7
• Circuit topology________________________________________7
LINEAR TRANSFORMATION
➢ HISTORY OF LINEAR TRANSFORMATION____________________7
• Early years____________________________________________8
• In the 90’s_____________________________________________8
➢ APPLICATIONS OF LINEAR TRANSFORMATION_______________9
• Linear transformation and machine language___________________9
• Application of linear transformation in deformation measurement__9
• Numerical calculation____________________________________9
• Calculus_______________________________________________9
• Basic computer graphic___________________________________9
• Cryptography__________________________________________10
➢ REFERENCES______________________________________________10
VECTOR SPACE
A vector space (also called a linear space) is a collection of objects called vectors, which may
be added together and multiplied (“scaled”) by numbers, called scalars. Scalars are often taken to
be real numbers, but there are also vector spaces with scalar multiplication by complex
numbers, rational numbers, or generally any field.
HISTORY OF VECTOR SPACES
Early history:
Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three-
dimensional space. Around 1636, French mathematicians René Descartes and Pierre de
Fermat founded analytic geometry by identifying solutions to an equation of two variables with
points on a plane curve. To achieve geometric solutions without using
coordinates, Bolzano introduced, in 1804, certain operations on points, lines and planes, which are
predecessors of vectors. This work was made use of in the conception of barycentric
coordinates by Möbius in 1827. The foundation of the definition of vectors was Bellavitis’ notion
of the bipoint, an oriented segment one of whose ends is the origin and the other one a target.
Vectors were reconsidered with the presentation of complex
numbers by Argand and Hamilton and the inception of quaternions by the latter. They are
elements in R2 and R4; treating them using linear combinations goes back to Laguerre in 1867,
who also defined systems of linear equations. In 1857, Cayley introduced the matrix
