Let V be an ndimensional real vector space and let T be a li

Let V be an n-dimensional (real) vector space and let T be a, linear transformation from V into V such that the range and null space of T are identical. Prove that n, the dimension of V is an even integer.

Solution

T: V->V is linear.

and Range (T)= Null (T).................................(1)

Recall the fact : (see proof below)

Dimension (V) = dimension Range(T) + Dimension Null(T)

                       = 2 dimension Range(T) from (1)

It follows that Dim(V) is an even integer

Dimension (V) = dimension Range(T) + Dimension Null(T)

From first homomorphism theorem

V/Null(T) is isomorphic to Range(T).

Thus Dim(V)-dim Null(T)= Dim Range (T).

Hence Dim(V)= dim Null(T)+ Dim Range (T).