Question

define range and null space of linear transformation​

Answer 1

Answer:

��:�� → �� is linear. The Null space of T, denoted N(T), is given by ��(��) = {�� ∈ ��|⁡��(��) = 0}. The Range of T, denoted R(T), is given by ��(��) = {��(��)|⁡�� ∈ ��}. Theorem 2.1: Suppose V and W are vector spaces over F, and ��:�� → �� is linear.

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 T : U → V is called a linear transformation or homomorphism of U into V if :

• T(x + y) = T(x) + T(y) ∀ x , y ∈ U
• T(ax) = aT(x) ∀ x ∈ U , a ∈ F

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

Range and null space of linear transformation :

Let U(F) and V(F) be two vector spaces and let T be a linear transformation from U into V , then the image or the range of T is denoted by R(T) (or Im(T) or T(U)) is the set of all vectors y ∈ V such that T(x) = y , x ∈ U .

i.e. R(T) = {y ∈ V : T(x) = y , x ∈ U}

And the null space of T is denoted by N(T) is the set of all the vectors x ∈ U , such that T(x) = 0' where 0' is the zero element of V .

i.e. N(T) = {x ∈ U : T(x) = 0' ∈ V is the zero element}

But if we regard the linear transformation T from U to V as a vector space homomorphism of U into V , then the null space of T is also called the kernal of T and it will be denoted by Ker(T) .

♦ R(T) is a subspace of V and N(T) is a subspace of U .