Internal Composition, External Composition – Complete Guide with Examples

 Introduction :-

Linear Algebra mathematics ka foundation mana jata hai. Agar aap CSIR NET, GATE, IIT JAM, M.Sc. Mathematics ya B.Sc. Mathematics ki preparation kar rahe hain, to Internal Composition, External Composition, aur Vector Space ko achhi tarah samajhna bahut zaroori hai.

Is article me hum in concepts ko simple language, real-life examples aur funny illustrations ke through detail me samjhenge.

What is Internal Composition?

अगर किसी non empty set के दो elements को combine करके result फिर से उसी set मे आ जाए तो उसे Internal Composition कहते है  

In Mathematically, 

Let V be a non empty set . An Internal composition on V is a mapping ❋: V X V → V such that for every u, v ∈ V,  u ❋ v ∈ V.

Example :- Let Z is a set of Integers and addition is operation then 2, 3  ∈ z and 2 + 3 = 5  ∈ Z. Hence it is a internal composition.

Example :- Let N is a set of Natural Numbers and Subtraction is operation then 3, 4  ∈ N but 3 - 4 = -1∉ N , Hence it is not Internal Composition.

What is External Composition?

External Composition मे एक scalar और एक vector को combine किया जाता है 

In Mathematically, 

Let F be a field and V be a non empty set . An external composition on V over F is a mapping ❋: F X V → V defined by α.v ∈ V where α ∈ F , v ∈ V.

Example :- Let V = R² = {(x,y) │x, y ∈ R } and F = R then take V = (1, 2) and F = -3

-3.(1, 2) = (-3 , -6) ∈ R², Hence it is a External Composition.

Example :- If V = R and F = C (Set of Complex Numbers) then it is not a External Composition.