Linear algebra for datascience
I’m writing this artcle to share knowledge to datascience enthusiast who are at early stage of their datascience life journey . I wish all of them joyfull learning experience.
In this article and coming articles we will learn few important concepts of linear algebra.So,don’t be unerved and relax yourself belive me its is not as difficult as it sounds….I promise you that this wont be as boring as or high school mathematics class lectures .
Welcome to the world of linear algebra folks !!
This document will review the fundamental ideas of linear algebra. We will learn about matrices, matrix operations,vectors, vector operations ,Eigen vector ,Eigen values ,linear transformations .
The tools of linear algebra open the gateway to the study of more advanced mathematics. A lot of knowledge buzz awaits you if you choose to follow the path of understanding, instead of trying to memorize a bunch of formulas.
INTRODUCTION:
The branch of mathematics that deals with the theory of systems of linear equations, matrices, vector spaces, determinants, and linear transformations.
ohh, dont worry these words might sound new to you we will learn what they are and why we use them ….cool okay!!
To begin with its imperative to know what is scalar quantity and vector quantity
Scalar quatities: Scalar quantity is defined as the physical quantity with magnitude and no direction.
eg: Mass, Duration,Length,Temperature,Charge,work,Distance,Energy..etc
Vecoter:A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.
eg: Force,Velocity,Accelration,Momentum , weight,Drag, Displacement ..etc
You are creating vector in your every activity!
Linear algebra is the math of vectors and matrices. Let n be a positive integer and let R denote the set of real numbers, then R n is the set of all n-tuples of real numbers. A vector ~v ∈ R n is an n-tuple of real numbers. The notation “∈S” is read “element of S.” For example, consider a vector that has three components:
v = (v1, v2, v3) ∈ (R, R, R) ≡ R 3
Vector operations: We now define the math operations for vectors. The operations we can perform on vectors ~u = (u1, u2, u3) and ~v = (v1, v2, v3) are: addition, subtraction, scaling, magnitude`(length), dot product, and cross product:
Vector addition :Given two vectors a and b, we form their sum a+b, as follows. We translate the vector b until its tail coincides with the head of a. (Recall such translation does not change a vector.) Then, the directed line segment from the tail of a to the head of b is the vector a+b.
Mathematically:
Addition of vectors satisfies two important properties.
The commutative law, which states the order of addition doesn’t matter:
a+b=b+a
The associative law, which states that the sum of three vectors does not depend on which pair of vectors is added first: (a+b)+c=a+(b+c)
Vector subtraction: Element wise subtraction of vector a and b.
Scaling :When we multiply a vector by a scalar it is called “scaling” a vector, because we change how big or small the vector is.
It is like changing the scale of a picture; the objects expand or shrink, but the directions remain the same.
Magnitude of vector:
The magnitude of a vector is the length of the vector. The magnitude of the vector a is denoted as ∥a∥.
To recall, vectors are multiplied using two methods
- scalar product of vectors or dot product
- vector product of vectors or cross product
Dot product: Dot product of two vectors produces scalar.
We can express the scalar product as:
a.b=|a||b| cosθ
where |a| and |b| represent the magnitude of the vectors a and b while cos θ denotes the cosine of the angle between both the vectors and a.b indicate the dot product of the two vectors.
Cross product of vector :
Cross product of two vector product vector .
The cross product is also expressed as |a||b|sinθ
Congratulations we have successfully learnt vector operations.
Thanks for reading this article .We will learn matrices, matrix operations,Eigen vector ,Eigen values ,linear transformations,Machine learning ,Deeplearning and computer vision concepts in upcoming articles.
Hit clap if you found this article useful and also comment below for any clarification and suggestions .See you all in next article ….
