Introduction:
In this question, we are given a vector space V and a linear operator T on V. We are also given that W is a subspace of V. We need to determine the condition under which W is invariant under T.

Definition of Invariant Subspace:
A subspace W of a vector space V is said to be invariant under a linear operator T if and only if for every vector w in W, the image of w under T is also in W. In other words, if T(w) is in W for all w in W, then W is invariant under T.

Condition for Invariance:
To determine the condition for W to be invariant under T, we need to consider the given statement "acT implies A Tα=0 B TaeW C D Ta=a None of these".

Analysis of the Given Statement:
Let's analyze each option given in the statement and determine its validity.

Option A: Tα=0
If Tα=0 for every vector α in W, then it implies that the image of every vector in W under T is the zero vector. In this case, W will be invariant under T because the zero vector is always in W.

Option B: TaeW
If T(a) is in W for every vector a in W, then it implies that the image of every vector in W under T is also in W. In this case, W will be invariant under T.

Option C: Ta=a
If Ta=a for every vector a in W, then it implies that the image of every vector in W under T is the same vector. In general, this condition does not guarantee that W is invariant under T. There may exist vectors in W whose images are not in W.

Option D: None of these
If none of the above conditions hold, then W may or may not be invariant under T. It depends on the specific properties of T and W.

Conclusion:
From the analysis above, we can conclude that option B (TaeW) is the correct condition for W to be invariant under T. If T(a) is in W for every vector a in W, then W is invariant under T.