n-Dimensional Euclidean Space as a Vector Space

UNDERSTANDING

It can be understood as a edge of a desk extending indefinitely in both the directions.

1d space

GEOMETRIC VISUALIZATION

Can it be visualized geometrically?

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MATH MODEL

R = set of real numbers.

Operations:

1. Addition: x + y ; x , y ∈ R

2. Scalar Multiplication: α.x ; x ∈ R

UNDERSTANDING

It can be understood by the top of the table or the surface of the black board extending indefinitely in all the directions.

2d space

GEOMETRIC VISUALIZATION

Can it be visualized geometrically?

  /

MATH MODEL

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


UNDERSTANDING

It can be understood as a room extending indefinitely in all the directions.

3d space

GEOMETRIC VISUALIZATION

Can it be visualized geometrically?

  /

MATH MODEL

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


GEOMETRIC VISUALIZATION

Can it be visualized geometrically?

  /

MATH MODEL

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.