Suppose T R10 rightarrow R10 is a linear map If dimNT 8 the

Suppose T: R^10 rightarrow R^10 is a linear map. If dim(N(T) = 8, then find dim(R(T)). If dim(R(T) = 10, explain why the transformation is one to one.

Solution

A)dim (R¹⁰)=dim (R(T))+dim (N(T))

=>10=8+dim (N(T))

=>dim (N(T))=2

B)

dim (R¹⁰)=dim (R(T))+dim (N(T))

=>10=10+dim (N(T))

=>dim (N(T))=0

HENCE ZERO ELEMENT IN R¹⁰ MAPS  ZERO ELEMENT IN R¹⁰

^HENCE EVERY ELEMENT MAPS ONE-ONE