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What is a nonconstant linear function?

  1. Apr 4, 2012 #1
    arent linear functions always constant?
     
  2. jcsd
  3. Apr 4, 2012 #2
    No, they aren't. Concider for example f(x)=x.
     
  4. Apr 4, 2012 #3

    Bacle2

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    Actually, the only constant function that is linear is the 0 function.

    If you have a linear map T:V-->W between V.Spaces (this generalizes to rings, etc.)

    then, if T(v)==wo , i.e., T(v)=wo for all v in V, then:

    T(v+v')=wo≠ T(v)+T(v')=wo+wo=2wo.

    A similar argument applies to maps from a vector space to its base field.
     
  5. Apr 5, 2012 #4
    well then i dont know what a constant and nonconstant linear functions are. Because f(x)=x is linear when graphed, so i was assuming linear is synonymous the word constant. as in a constant rate of change or constant slope.
     
  6. Apr 5, 2012 #5

    Office_Shredder

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    A constant function is a function which always takes the same value, for example f(x)=2.
    All linear functions on Rncan be written as y=Ax where A is a matrix (in one dimension, just a number)
     
  7. Apr 5, 2012 #6

    Bacle2

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    spoke:

    You may be confusing constant rate of change, i.e., constant derivative--a property of linear functions-- with constant function.
     
  8. Apr 5, 2012 #7
    So would this relation be an example constant function? {(1,2), (2,2), (3,2), (4,2)}
     
  9. Apr 6, 2012 #8

    Bacle2

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    Yes, exactly, that is what a constant function is like when seen as a subset of AxB.

    Not to nitpick, but you may want to specify the sets A,B where you are defining

    your function as a subset of AxB; here, A is clearly specified, but it is not clear

    what B is (unless you assume your function is onto B).
     
  10. Apr 6, 2012 #9
    A linear function is constant if and only if its slope is zero. By contaposition, a linear function is not constant (i.e. non-constant) iff its slope is different from zero.
     
  11. Apr 6, 2012 #10

    Bacle2

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    Your right, Dickfore, but your example is that of a map from ℝ to itself may be too
    specific for a general definition of function.
     
  12. Apr 6, 2012 #11
    OK, make
    [tex]
    \mathbf{y}_{n \times 1} = \hat{A}_{n \times m} \cdot \mathbf{x}_{m \times 1} + \mathbf{b}_{n \times 1}
    [/tex]
    This is a general mapping from [itex]\mathbb{C}^m \rightarrow \mathbb{C}^n[/itex]. But, now, the function may be constant in a more general case, when [itex]\mathrm{rank}A \le m < n[/itex].
     
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