Question

Let V and W be vector spaces over a field F and let V' be a proper subspace of V. Are { T belongs to Hom(V,W)|Ker(T) is contained in V'} and {T belongs to Hom (V,W)|U is contained in Ker(T)} subspaces of Hom(V,W)?

Answer 1

i get this mixed up
A|B is A given B or B given A

Answer 2

It is A given B. But how is that relevant to my question?

Answer 3

what is up with U is contained in Ker(T), where is U coming from

Answer 4

it is supposed to be V' sorry

Answer 5

Just give me some starting insight so that I can do this by myself

Answer 6

yes

Answer 7

theyre all still subspaces of V

Answer 8

But how do I prove that?

Answer 9

if ket(T) c V' and V' c V then ker(T) c V

Answer 10

V' c Ker(T)
Ker(T) c T
T is hom(V,W)
so (v1+v2) T = v1T+v2T