Don't understand this simple vector space problem

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SUMMARY

The discussion centers on the relationship between the null spaces of linear operators S and T in the context of their composition ST being invertible. It is established that if ST is invertible, then null(ST) is zero, which implies that null(S) must also be a subspace of the zero vector space. The confusion arises from the assumption that null(S) could contain elements not in null(ST), which is incorrect given the invertibility of ST. The key takeaway is that if ST is invertible, then S must be injective.

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  • Basic comprehension of vector space composition
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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 ?
 
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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).
 
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 :(
 
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?
 
it depends which convention on composition is being used. some people write ST for first S then T, but not me.
 

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