Vector Spaces and Linear Transformation

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.

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