Let T Rn rightarrow Rm be a linear transformation What is th

Let T: R^n rightarrow R^m be a linear transformation, What is the dimension of the range of T if T is one-to-one? Explain. What is the dimension of the kernel of T if T maps R^n onto R^m ? Explain

Solution

a)

By rank nullity theorem

rank(T)+nullity(T)=n

If T is one to one then nullity(T)=0

Hence, rank(T)=dim range (T)=n

b)

T is onto so, rank(T)=m

Case 1: m>n

rank(T)+nullity(T)=n

m+nulllity(T)=n

nullity(T)=n-m<0

which is not possible

Case 2: m<=n

nullity(T)=n-m

ie dim kernel(T)=n-m