LetS:V?UandT:U?Wbe linear transformations between vector spacesU,VandW.

1. a)Show that ker(S)?ker(T?S).
2. b)Show that im(T?S)?im(T).
3. c)Show that im(S)?ker(T) if and only ifT?Sis the zero mapping (i.e.,T(S(v)) = 0Wfor allv?V).
4. d)For each of (a), (b) and (c), find an example of a specific pair of linear transformationsSandTwith source and targetR2wherestrictinequality holds.[Hint:Keep it simple!]

Let S : V - U and T : U - W be linear transformations between vector spaces U, V and W. a) Show that ker(S) C ker(T o S). b Show that im(To S) Cim(T). C Show that im(S) C ker(T) if and only if To S is the zero mapping (i.e., T(S(v)) = Ow for all v E V). d) For each of (a), (b) and (c), find an example of a specific pair of linear transformations S' and T with source and target R2 where strict inequality holds. [ HINT: Keep it simple!]