Question

kernal of a linear transformation

Answer 1

Answer:-

Let V and W be vector spaces over some field K. Now, let ϕ:V⟶W be a linear mapping/transformation between the two vector spaces. We define the kernel of ϕ to be the set ker(ϕ)={v∈V|L(v)=0}. So the kernel of ϕ is the set of all elements of V that get taken to the zero vector by ϕ. ker(ϕ) is a subvectorspace of V. On the other hand, we define the image of ϕ to be the set im(ϕ)={w∈W|∃v∈Vs.tϕ(v)=w}. So the image of ϕ is the set of all elements in W that are mapped to by ϕ. im(ϕ) is a subvectorspace of W.

If you’ve taken an abstract algebra course and are familiar with the notion of a quotient group, then for a linear transformation, im(ϕ)≅V/ker(ϕ) which means that the image of ϕ is isomorphic to the quotient of V by ker(ϕ). Since dim(V/ker(ϕ))=dim(V)−dim(ker(ϕ)), then we see that dim(im(ϕ))=dim(V)−dim(ker(ϕ)) or dim(im(ϕ))+dim(ker(ϕ))=dim(V) where dim is the dimension of the vector space.

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 .

Kernal of a linear transformation :

Let f be a linear transformation of a vector space U(F) into a vector space V(F) , then the set of all those elements of U which are mapped into zero element of V is called kernal of the linear transformation f .

♦ Kernal of the linear transformation f is denoted by Ker(f) . Thus , Ker(f) = {x ∈ U : f(x) = 0' , where 0' is the zero vector of V}