## Vector Space-

A set V is called vector space on which two operations are defined.

## 1- Vector Addition

## 2- Scalar Multiplication

Further, vector addition must follow the following properties and laws for vector addition.

## 1- Closure Property

If u and v are two vectors in V then u+v must be in V.

## 2- Commutative Law

u+v must also be equal to v+u i.e **u+v = v+u**

## 3- Associative Law

If there are three vectors u, v and w then

**(u + v) + w = u + (v + w)**

## 4- Existence of Additive Identity

**V** must have additive identity 0 for which v+0= v and 0+v=v

## 5- Existence of Additive Inverse

There must be additive inverse -v of v such that v + (-v) = 0

In addition, vector addition must follow the following properties and laws for scalar multiplication.

## 6- Closure Property-

If x in F is a scalar and v is a vector in V, then x.v must be in V.

**7- Distributive Law-**

If x is a scalar in field F and u and v are vectors in V, then

x .** (u + v) = x.u + x.v**

**8- Associative Law**

If x, y are in field F and v is in V then

**(**xy) .** v = x. (y.v)**

**9-Existence of Multiplicative Identity**

If v is a vector in V and 1 is identity in F then

v.1 = 1.v = v

**10- Existence of Multiplicative Inverse**

If v, u are vectors in V and, the u is called multilicative inverse of of v

if v.u=u.v=1

## Linear Transformation

If U and V are two vector spaces over vector field F and T is a transformation from U to V i.e

T: U->V. T is called linear transformation if it follows the following properties.

**1- T(u+v) = T(u) + T(v)**

**2- T(xv) = x T(v)**

Where u and v are vectors and x is a scalar.

**Note-** This topic is very important in machine learning and data science in field of data transformation and preprocessing techniques.