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Math Help - Isomorphism proves: f and g are isomorphisms, prove f + g is an isomorphism

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    Isomorphism proves: f and g are isomorphisms, prove f + g is an isomorphism

    Question:

    Suppose f and g are isomorphisms from U to V. Prove or disprove each of the following statements:
    a) The mapping f + g is an isomorphism from U to V

    I have no idea where to start. Do I need to show that f(u) + g(u) = (f+g)(u)? Do I need to show that (f+g) is 1-1 and onto?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by zeion View Post
    Question:

    Suppose f and g are isomorphisms from U to V. Prove or disprove each of the following statements:
    a) The mapping f + g is an isomorphism from U to V

    I have no idea where to start. Do I need to show that f(u) + g(u) = (f+g)(u)?
    no, that's true by definition.

    Do I need to show that (f+g) is 1-1 and onto?
    yes, and one more thing. (f + g)(xy) = (f + g)(x) \cdot (f + g)(y)

    i would start with this one first. what can you find?
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    Quote Originally Posted by Jhevon View Post
    no, that's true by definition.

    yes, and one more thing. (f + g)(xy) = (f + g)(x) \cdot (f + g)(y)

    i would start with this one first. what can you find?
    Ok let's try...

    Let A be a basis for U and B be a basis for V, and let v belong to V and u belong to V , then
     v = {a_1v_1+...+a_nv_n} , u = {a_1u_1+...+a_nu_n}
    (f + g)(vu) = (f + g)[(a_1v_1+...+a_nv_n)(a_1u_1+...+a_nu_n)]
    = (f + g)(a_1+...+a_n)(v_1u_1+...+v_nu_n)
    Does that look right?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by zeion View Post
    Ok let's try...

    Let A be a basis for U and B be a basis for V, and let v belong to V and u belong to V , then
     v = {a_1v_1+...+a_nv_n} , u = {a_1u_1+...+a_nu_n}
    (f + g)(vu) = (f + g)[(a_1v_1+...+a_nv_n)(a_1u_1+...+a_nu_n)]
    = (f + g)(a_1+...+a_n)(v_1u_1+...+v_nu_n)
    Does that look right?
    that's not what you were supposed to show. and why are you choosing bases? I suppose this is a linear algebra problem as opposed to an abstract algebra problem(?) if so, then we would use the definition that an isomorphism between two vector spaces is a bijective linear transformation from one to the other. so, is it true that if we take two such transformations, their sum will be such a transformation?
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    Quote Originally Posted by zeion View Post
    Question:

    Suppose f and g are isomorphisms from U to V. Prove or disprove each of the following statements:
    a) The mapping f + g is an isomorphism from U to V

    I have no idea where to start. Do I need to show that f(u) + g(u) = (f+g)(u)? Do I need to show that (f+g) is 1-1 and onto?
    What kind of structures are U and V? Groups? Rings? Fields? Vector spaces?
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