It can be understood by an edge of a desk extending indefinitely in both the directions.
R = set of real numbers
Operations:
1. Addition: x + y ; x , y ∈ R
2. Scalar Multiplication: α.x ; x ∈ R
It can be understood by the top of a table or the surface of a black board extending indefinitely in all the directions.
R2 = {(x, y) : x, y ∈ R}
Operations:
1. Addition: (x1 , x2 ) + (y1 , y2) = (x1 + y1 , x2 + y2) ; x1 , x2 , y1 , y2 ∈ R
2. Scalar Multiplication: α . (x1 , y1) = (α x1 , α y1) ; x1 , y1 ∈ R
It can be understood by the inside of a room extending indefinitely in all the directions.
R3 = {(x, y, z) : x, y, z ∈ R}
Operations:
1. Addition: (x1 , x2 , x3) + (y1 , y2 , y3) = (x1 + y1 , x2 + y2 , x3 + y3) ; x1 , x2 , x3 , y1 , y2 , y3 ∈ R
2. Scalar Multiplication: α . (x1 , y1 , z1) = (α x1 , α y1 , α z1) ; x1 , y1 , z1 ∈ R
Rn = {(x1 , x2 , x3 , ... , xn) : x1 , x2 , x3 , ... , xn ∈ R}
Operations:
1. Addition: (x1 , x2 , x3 , ... , xn) + (y1 , y2 , y3) , ... , yn) = (x1 + y1 , x2 + y2 , x3 + y3 + ... + xn + yn ) ; x1 , x2 , x3 , ... , xn , y1 , y2 , y3 , ... , yn ∈ R
2. Scalar Multiplication: α . (x1 , x2 , x3 , ... , xn) = (αx1 , αx2 , αx3 , ... , αxn) ; x1 , x2 , x3 , ... , xn ∈ R
System (Rn, +, .) together with R, denoted by Rn, is a vector space over R, known as the n-dimensional Euclidean space.