Answer to Question #207918 in Linear Algebra for Simphiwe Dlamini

Question

Prove that there does not exist a linear map T : R5 - R5 such that range T = null T.

Answer 1

The given function is $T:R^5\rightarrow R^5$

Range T = null T

As per the rank nullity theorem,

If $T:V\rightarrow W$ is a linear map, then dim(Range T) + dim (null T)= dim V

So, dim(Range T)=dim(null T) and dim(V)=5

Now, applying the nullity theorem,

dim(Range T)+ dim (null T)= 5

dim(Range T)+ dim (Range T)=5

2 dim (Range T)=5

From the above equation, we can see that the left hand side is odd and the right hand side is even, which is the contradiction. Hence there is no existence of linear map.

$T:R^5\rightarrow R^5$ such that Range(T)=null T