Simple question on non singular linear transformation

[cocobaby]

Homework Statement

Given that "If T(Ta)=0, then Ta=0",
can we say that the linear transformation on V is nonsingular?

Homework Equations

The Attempt at a Solution

Since what the statement implies is that T has only zero subspace of V as its null space, can we not say that it's nonsingular?

 

[Mark44]

I don't think you can say that at all. Suppose T:V --> V is defined by T(x) = 0 for any x in V. The nullspace of T is all of V, so T is definitely noninvertible. What does that imply about T being singular or nonsingular?