Question

What is linear transformation?​

Answer 1

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (for example, two vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication

Answer 2

Vector space :

(V , +) be an algebraic structure and (F , + , •) be a field , then V is called a vector space over the field F if the following conditions hold :

1. (V , +) is an abelian group .
2. ku ∈ V ∀ u ∈ V and k ∈ F
3. k(u + v) = ku + kv ∀ u , v ∈ V and k ∈ F .
4. (a + b)u = au + bu ∀ u ∈ V and a , b ∈ F .
5. (ab)u = a(bu) ∀ u ∈ V and a , b ∈ F .
6. 1u = u ∀ u ∈ V where 1 ∈ F is the unity .

♦ Elements of V are called vectors and the lements of F are called scalars .

♦ If V is a vector space over the field F then it is denoted by V(F) .

Vector space homomorphism or linear transformation :

Let U(F) and V(F) be two vector spaces over the same field F , then a mapping f : U → V is called a linear transformation or homomorphism of U into V if :

• f(x + y) = f(x) + f(y) ∀ x , y ∈ U
• f(ax) = af(x) ∀ x ∈ U , a ∈ F

In another words , a mapping f : U(F) → V(F) is called a linear transformation or homomorphism of U into V if : f(ax + by) = af(x) + bf(y) ∀ x , y ∈ U , a , b ∈ F .

Example :

The mapping f : R² → R² defined as f(x , y) = (x , 0) is a linear transformation .

Let X = (x , y) and Y = (x' , y') ∈ R² and a , b ∈ R .

Then , f(X) = f(x , y) = (x , 0)

f(Y) = f(x' , y') = (x' , 0)

Now ,

aX + bY = a(x , y) + b(x' , y')

= (ax , ay) + (bx' , by')

= (ax + bx' , ay + by')

Also ,

f(aX + bY) = f(ax + bx' , ay + by')

= (ax + bx' , 0)

= (ax , 0) + (bx' , 0)

= a(x , 0) + b(x' , 0)

= af(X) + bf(Y)

Hence , f is a linear transformation .