Question

Let T₁​:V→V and T₂​:V→V be two linear transformations with dim(V)=n. suppose dim(Range(T1​))= m≥1 and T₁ ​(T₂​(v))=0∀v∈V. What can you say about the dimension of Range (T₂​) ?

Answer 1

The dimension of Range (T₂​) is n - m.

Step-by-step explanation:

Given two linear transformations T₁​:V→V and T₂​:V→V, with dim(V)=n, and dim(Range(T1​))= m≥1, and T₁ ​(T₂​(v))=0∀v∈V, we need to determine the dimension of Range (T₂​).The condition T₁ ​(T₂​(v))=0∀v∈V implies that the range of T₂​ lies entirely in the null space (kernel) of T₁​. Since the dimension of the null space of T₁​ is n - m (where n is the dimension of V and m is the dimension of Range(T₁​)), the dimension of Range(T₂​) is n - m.Therefore, the dimension of Range (T₂​) is n - m.