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Math Help - Linear Algebra: Linear Transformations

  1. #1
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    Unhappy Linear Algebra: Linear Transformations

    I am working on several linear algebra problems and found myself stuck on two problems. Any helpful hints or suggestions on how to go about solving them will begreatly appreciated.

    1) Given a 3 x 5 matrix and T(x)=Ax. Find range(T).
    How do I find the range?

    2) Let V be an inner product spae with a subspace W having B={w1,w2,...,wn} as an orthonormal basis. Show that the T(v)= the sum i=1 to n of <v,wi>wi is a linear transformation.
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  2. #2
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    Quote Originally Posted by mathwiz2006 View Post
    1) Given a 3 x 5 matrix and T(x)=Ax. Find range(T).
    How do I find the range?
    It depends on the matrix A.

    2) Let V be an inner product spae with a subspace W having B={w1,w2,...,wn} as an orthonormal basis. Show that the T(v)= the sum i=1 to n of <v,wi>wi is a linear transformation.
    Remember you need to show that T(k\bold{x}) = kT(\bold{x}) and T(\bold{x}+\bold{y}) = T(\bold{x}) + T(\bold{y}).
    Can you show that?
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  3. #3
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    1) A= [3,-2,6,-1,15; 4,3,8,10,-14; 2,-5,1,-4,20]

    2) I know that I must show
    and .
    but I am mostly confused because it is the sum of the given and am having trouble showing that they work
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