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Math Help - Linear Algebra Proofs regarding Linear maps, null spaces, and composition.

  1. #1
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    Linear Algebra Proofs regarding Linear maps, null spaces, and composition.

    Let U, V and W be finite-dimensional vector spaces over F and S is in L(V,W) and T is in L(V,W). Show that:
    nullity(composite of f and g) is less than or equal to nullity(T)+nullity(S).

    Thank you.
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  2. #2
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    Re: Linear Algebra Proofs regarding Linear maps, null spaces, and composition.

    Frankly, you seem to have very little interest in these problems- you are not even copying them correctly! (Or your text book is just really really bad.) You cannot talk about "nullity of f and g" when there are NO functions called "f" or "g" in the problem. Since you do talk about linear functions S and T, I would think that you meant "the composition of S and T" but that still makes no sense. S is in L(V, W) so S maps vectors in V to vectors in W. T is also in L(V, W) so, again, maps vectors in V to W. To be able to compose ST, we would have to have T in L(V, W) and S in L(W, Z) for some vector space Z. To be able to compose TS, we would have to have S in L(V, W) and T in L(W, Z).

    If you meant "S is in L(V, W) and T is in L(W, Z), show that nullity(TS) is less than or equal to nullity(T)+ nullity(S)", then take v in nullity(S). What is S(v)? What is TS(v)?
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  3. #3
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    Re: Linear Algebra Proofs regarding Linear maps, null spaces, and composition.

    Oh whoops my bad. I'm a bit out it. here, let me copy it correctly.

    Let U, V and W be finite-dimensional vector spaces over F and S is in L(U,V) and T is in L(V,W). Show that:
    nullity(composite of T and S) is less than or equal to nullity(T)+nullity(S).

    So yes, you had the right idea in interpretting my mis-copied content.

    I'm not quite sure, but i had the idea of starting the proof off with stating that since null(S) is a subspace of U and null(T) is a subspace of T, null(S) has a basis (u1, u2, ..., un), and null(T) has a basis (v1, v2, ..., vm), thus implying that nullity(S)=n and nullity(T)=m. However, i'm not quite sure what the relation is between compositions and nullities...

    thank you for the response.
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  4. #4
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    Re: Linear Algebra Proofs regarding Linear maps, null spaces, and composition.

    *Correction, i'm not quite sure what the relation is between compositions, nullities, and dimensions, meaning i don't understand how to put these ideas together. I understand that the composition is T(S(u)) given that u is in U. But from that point on, i'm not sure what to do or how to think about the problem.
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