Let T is a linear transformation from Rn rightarrow Rm. Prove that Im(T) is a su
ID: 2939095 • Letter: L
Question
Let T is a linear transformation from Rn rightarrow Rm. Prove that Im(T) is a subspace of Rm.Explanation / Answer
to prove Im (T) is a subspace of R m , it is enough toshow the necessary and sufficient condition to be a subspace. to show any condition on any set, it is our primary duty to checkwhether the said set is empty or non empty. on empty sets imposing any thing is useless. keeping this in view, we know that every lineartransformation is a homomorphism on vector spaces. so, when 0 vector is in Rn and T : Rn-->Rm is a linear transformation , T(0) must be in Im(T). so, Im (T) is a non empty set. now, suppose a , b are any scalars and T(u) ,T(v) are any vectors in Im(T). observe that Im (T) means the images of the vectors ofRn under the influence of T which are inRm. so, they look like T( ). consider a T(u) +b T(v). we are required to show this linear combination is also in Im(T). T(u) , T(v) are in Im (T) means u , v are inRn. so, a u +b v is in Rn . since Rn is a vector space. linear combination of vectors is also a vector . so, when a u+b v is a vector in Rn, the correspondingimage of it under T must be in Im (T). i.e. T(a u+b v) is in Im (T). but T is a linear transformation. so, T(a u +b v) = a T(u) +b T(v)is in Im(T). thus, we have shown that when ever a, b are scalars and T(u),T(v) are arbitrary vectors in Im (T), their linear combination isalso a vector in Im (T). necessary and sufficient condition to be a subspace issatisfied. thus , Im (T) is a subspace. since T is a liner transformation , this linear combination can bewritten as
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.