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Math Help - Question on linear transformation in analysis

  1. #1
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    Question on linear transformation in analysis

    Hi everybody
    I am stuck on this question and hope to get some ideas from u guys
    (a) If T: R^m --> R^n is a linear transformation, show that there is a number M such that |T(h)|=< M|h| for h in R^m
    (b) Prove that T is continuous

    Here is my attempt:
    T: R^m --> R^n is a linear transformation if:

    T(x+y) = T(x)+T(y) for all x, y in R^m
    T(kx) = kT(x) for k is real number

    What do I have to do now? and how do you start part b?

    Thanks for any suggestions
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by knguyen2005 View Post
    Hi everybody
    I am stuck on this question and hope to get some ideas from u guys
    (a) If T: R^m --> R^n is a linear transformation, show that there is a number M such that |T(h)|=< M|h| for h in R^m
    (b) Prove that T is continuous

    Here is my attempt:
    T: R^m --> R^n is a linear transformation if:

    T(x+y) = T(x)+T(y) for all x, y in R^m
    T(kx) = kT(x) for k is real number

    What do I have to do now? and how do you start part b?

    Thanks for any suggestions
    Let e_{k=1,\ldots, m} be a base of \mathbb{R}^m and let h=\sum_{k=1}^m h_k e_k. Then you have (by linearity of T, the triangle inequality, and because |h_k|\leq |h| for all k=1,\ldots, m):
    |T(h)| = |\sum_{k=1}^m h_k T(e_k)|\leq \sum_{k=1}^m |h_k| \,|T(e_k)|\leq |h| \, \sum_{k=1}^m|T(e_k)|= M \, |h|
    where M := \sum_{k=1}^m|T(e_k)|
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