A subspace is also a vector space, why?

[HelpNeeder]

I do not understand how we can be sure that axioms 2, 3 and 7-10 are automatically true in H. If that is the case, then axioms 1, 4 and 6 also need to be automatically true because they apply to all element of V, including those in H?
So what I do not understand is why we only have to check if these three properties are true and not the other 7?

Thank you in advance.

 

[lex]

I do not understand how we can be sure that axioms 2, 3 and 7-10 are automatically true in H. If that is the case, then axioms 1, 4 and 6 also need to be automatically true because they apply to all element of V, including those in H?
So what I do not understand is why we only have to check if these three properties are true and not the other 7?
Thank you in advance.

For the 'subspace' H to be a vector space, we need H to be:

2, 3, 7-10 apply to H simply because the elements of H are elements of V and therefore already act in this manner, under these operations. They inherit these properties from V.
However 1, 4, 5, 6 apply restrictions which are not true in general for members of V. E.g. Read my axiom (1) above. It is NOT in general the case that the sum of any u and v members of V, will be in H.
In any case, we only have to verify these axioms for members of H. E.g. only for u and v members of H, do we need u+v to be in H.
As shown in the explanation in your post, these 4 axioms are true for members of H (and referring to H, not V) because of the defining properties (a) (b) (c)

 

[Dr.Peterson]

View attachment 26369
View attachment 26368
I do not understand how we can be sure that axioms 2, 3 and 7-10 are automatically true in H. If that is the case, then axioms 1, 4 and 6 also need to be automatically true because they apply to all element of V, including those in H?
So what I do not understand is why we only have to check if these three properties are true and not the other 7?

Thank you in advance.

Properties 2, 3, 7-10 are necessarily true, because they are properties of the operation itself, and we are using the same operation in the subspace.

Properties 1, 4, and 6 have to be separately proved, because they involve how the set H interacts with the operation; there are subsets for which these properties are not true. They are not "automatically" true, because they are not just properties of the operation or of individual elements, but of H as a whole.

Property 5 is implied by a, b, c together with the axioms of V, as explained.

But all of this is stated in what you quote. What are you missing? I suspect it is the fact I underlined.

 

[HelpNeeder]

Thank you very much. Indeed, the fact Dr. Peterson underlined is what I was missing.