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.

References Links

Share to Your Friend

Be the first to comment on "Vector Spaces and Linear Transformation"

Leave a comment