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Linearity - Definition

  1. Sep 27, 2011 #1
    Probably I'm just being stupid but:
    According to Wikipedia.

    * Additivity (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition.
    * Homogeneity of degree 1: f(αx) = αf(x) for all α.

    f[x_] := x + 2;
    f[3 + 7]
    f[3] + f[7]


    which indicates x+2 is not linear but obv it is a linear function. Where is the error?
  2. jcsd
  3. Sep 27, 2011 #2


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    Science Advisor

    There is no error. You are talking about two different things. The "linear function" defined in elementary algebra or PreCalculus is not the same as the "linear operators" or "linear functions" defined in linear algebra or abstract algebra.

    For one, thing, if you have a "linear function", f, from [itex]R^1[/itex] to [itex]R^2[/itex], its "image", the set of all points in [itex]R^2[/itex] that are given by f(t), would be a straight line through the origin. The graph of a linear equation in the "elementary" sense (the set of points in [itex]R^2[/itex] (x, f(x))) is a straight line not necessarily through the origin. In linear algebra, we would call the first a "linear subs-space" of [itex]R^2[/itex] and the second a "linear manifold".
  4. Sep 27, 2011 #3
    So how is linearity defined for high-school mathematics?

    I've done linear algebra and calculus(one and several variables) though. For a function
    f[x]=x+2, how is linearity defined.
  5. Sep 27, 2011 #4


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    As you said in your first post, linearity of a function or operator means the function/operator has the following properties:

    f(x + y) = f(x) + f(y),
    f(ax) = af(x)

    As HallsofIvy already mentioned, when calling a function such as "y = x+2" linear, the word linear is being used in a different sense than the above definition (as you already noticed that 'linear' polynomials with a non-zero constant don't satisfy the conditions of linearity!). The reason such functions are called 'linear' is because they are lines, which is the root of the word linear. You should be able to figure out from context which sense of the word is being used. If talking about mx + b type lines, it likely just means "lines". If talking about linear algebra or things like the derivative or integral being linear operations, it is referring to the definition at the start of the post.
    Last edited: Sep 27, 2011
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