Let U, V, and W be vector spaces over F where F is R or C. Let S: U -> V and T: V -> W be two linear maps. Recall that the composition TS is defined by (TS)(x) = T(S(x)).

(a) Prove that if TS is injective, then S is injective.

(b) Prove that if TS is surjective, then T is surjective.

(c) Assume that TS is bijective. Prove that S is surjective if and only if T is injective.

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(a) Assume that TS is injective. We need to prove that S is injective.

Injective means one-to-one => TS is one-to-one. This implies that TS(x) = 0 where x = 0.

Assume that S(x) = 0. We want to prove that x = 0.

S(x) = 0 => T(S(x)) = T(0) => T(0) = 0 => x = 0. So, S is one-to-one => S is injective.

is this okay?? should i write more?

(b) Assume that TS is surjective. We need to prove that T is surjective.

Surjective means onto => TS is onto W.

Assume R(TS) = W. We want to prove R(T) = W.

I have NO idea how to do this one...

(c) Assume TS is bijective. This means that TS is both injective and surjective => TS is one-to-one correspondence. We need to prove that S is surjective iff T is injective.

I know after proving (a) and (b), we can say that S is injective and T is surjective.

"=>" Assume S is surjective. We need to prove that T is injective.

TS is bijective => TS is injective => S is injective.

But we assumed that S is surjective which then implies that S is an isomorphism.

TS is bijective => TS is surjective => S is surjective.

dim(U) = N(S) + R(S) Because S is an isomorphism, N(S) = 0 and R(S) = dim(V). Thus, dim(U) = dim(V).

dim(U) = N(TS) + R(TS) Because TS is injective, N(TS) = 0 and R(TS) = dim(W). Thus, dim(U) = dim(W).

By Dimension Theorem, dim(V) = dim(W). Thus T is injective.

(Is this right?? can I just conclude that?)

"<=" Assume T is injective. We need to prove that S is surjective.

TS is bijective => TS is surjective => T is surjective. But we assumed that T is injective which implies that T is an isomorphism.

TS is bijective => TS is injective => T is injective.

dim(V) = N(T) + R(T) Because T is an isomorphism, N(T) = 0 and R(T) = dim(W). Thus, dim(V) = dim(W).

Now I'm really confused. I think I just mainly don't understand all this bijective and surjective stuff.

Thanks so much to those who help me with this problem. I really need it.