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Math Help - Linear Transformation Composition Proof

  1. #1
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    Linear Transformation Composition Proof

    I need help with this linear algebra proof.

    Show that the composition of a linear transformation is also a linear transformation. Let T: Rn to Rm and S: Rm to Rp be linear transformations. Show that S(T): Rn to Rp is also a linear transformation. Show that the composition respects scalar multiplication and vector addition.

    Thanks.
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  2. #2
    Member ModusPonens's Avatar
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    Re: Linear Transformation Composition Proof

    S(T(u+v))=S(T(u)+T(v))=S(T(u))+S(T(v))

    Now you can do the scalar multiplication.
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  3. #3
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    Re: Linear Transformation Composition Proof

    note that the composition is usually written ST, or SoT, not S(T) (S doesn't have "linear transformations" as its input, but the VECTORS T(v)).
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  4. #4
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    Re: Linear Transformation Composition Proof

    II. Show that the composition of a linear transformation is also a linear transformation. Let T: Rn Rm and S: Rm Rp be linear transformations. Show that (S o T)(x) = S(T(x)). Show that the composition respects scalar multiplication and vector addition.
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  5. #5
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    Re: Linear Transformation Composition Proof

    (SoT)(x) is by DEFINITION the function that takes x to S(T(x)):

    x-->T(x)-->S(T(x))

    so there's nothing to show. what there is to show is that:

    a)(SoT)(x+y) = (SoT)(x) + (SoT)(y) for all vectors x and y
    b) (SoT)(ax) = a(SoT(x)), for all scalars a, and all vectors x.

    ModusPonens already proved (a) in post #2, can you prove (b)?

    this is mostly "manipulating the definitions"...it sounds as if you are unsure of what it is you are being asked to prove. so, how do we know a function from one vector space to another is a linear transformation?
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