Don't understand this simple vector space problem

[octol]

Don't understand this reasoning with respect to linear operators.

Let S and T be linear operators on the finite dimensional vector space V. Then assuming the composition ST is invertible, we get
$$\text{null} \; S \subset \text{null} \; ST$$

Why is that? I thought hard about it but I simply cannot follow. Is it not possible to have an element x of V that is in the nullspace of S but not in the nullspace of ST ? i.e. S maps x to 0 but T maps x to y where S don't map y to 0 ?

 

[matt grime]

If ST is invertible, then the null space of it is zero. I think you ought to re-examine your question.

It is certainly true that null(T) is a subspace of null(ST).

 

[octol]

yes I understand that the null space of ST i zero, and that null(T) is a subspace of null(ST), but how can we say that null(S) is a subspace of null(ST) ? I must be doing some kind of faulty thinking :(

 

[matt grime]

You can't say null(S) is a subspace of null(ST), in general. It isn't. Trivially. Howver, you asserted that ST was invertible, and at no point attempted to use this fact. Thus null(ST)=0, so you're asking 'is null(S) a subspace of the zero vector space'. Well, what is the only subspace of the zero space? I.e. is S injective?

 

[mathwonk]

it depends which convention on composition is being used. some people write ST for first S then T, but not me.