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Thread: Confusion about linear transformations

  1. #1

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    Confusion about linear transformations

    Hi all,
    I am reading a Linear Algebra book.
    In the chapter called linear transformations, it says that a linear transformation is a function like T from one vector space to another when the following conditions are met:
    T(x + Y) = T(x) + T(y)
    T(ax) = aT(x)

    Now, that makes me think about the following function:
    f(x) = 3x + 7
    or
    y = 3x + 7

    This function definitely fails those two conditions, so, does it mean that this function f(x) = y = 3x + 7 is NOT linear !!!???

    In all my life in all the math books that I have read, and all the math classes that I have attended, I have read and heard that something like f(x) = y = 3x + 7 is indeed linear.
    But, now the linear algebra book is saying that it is not.

    What's going on in here?
    Am I missing anything?
    Can somebody please shed some light on this?

    Thanks.
    Ilia

  2. #2
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    Re: Confusion about linear transformations

    f(x) = 3x + 7 is a linear function of course, but not a linear transformation.

    For f(x) to be a linear transformation, both f(c*x) = c*f(x) and f(x) + f(y) = f(x+y) must be true.

    Take your f(x) and compute f(2*x) = 3*(2*x) + 7 = 6*x + 7
    But 2*f(x) = 2*(3*x + 7) = 6*x + 14, so it fails the first condition

    Now, take your f(x) and compute f(x) + f(y) = (3*x + 7) + (3*y + 7) = 3*x + 3*y + 14
    But f(x+y) = 3*(x+y) + 7 = 3*x + 3*y + 7, so it fails the second condition

  3. #3
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    Re: Confusion about linear transformations

    Quote Originally Posted by OptionBase1 View Post
    For f(x) to be a linear transformation, both f(c*x) = c*f(x) and f(x) + f(y) = f(x+y) must be true.
    Why both halves? It seems to me that f(c*x)=c*f(x) is a subset of f(x) + f(y) = f(x + y), because it's the case where x = y=1 and c=2. Is it possible for the more general case to be true while f(c*x) <> c*f(x)?
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  4. #4
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    Re: Confusion about linear transformations

    In the context of the question posed, I guess the proper notation would be in terms of u's and v's rather than x and y, where u and v are both vectors, and then c represents a scalar. So they aren't simply restating the same condition.

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