What is a nonconstant linear function?

  1. arent linear functions always constant?
  2. jcsd
  3. No, they aren't. Concider for example f(x)=x.
  4. Bacle2

    Bacle2 1,168
    Science Advisor

    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. 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. Office_Shredder

    Office_Shredder 4,487
    Staff Emeritus
    Science Advisor
    Gold Member

    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. Bacle2

    Bacle2 1,168
    Science Advisor


    You may be confusing constant rate of change, i.e., constant derivative--a property of linear functions-- with constant function.
  8. So would this relation be an example constant function? {(1,2), (2,2), (3,2), (4,2)}
  9. Bacle2

    Bacle2 1,168
    Science Advisor

    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. 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. Bacle2

    Bacle2 1,168
    Science Advisor

    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. OK, make
    \mathbf{y}_{n \times 1} = \hat{A}_{n \times m} \cdot \mathbf{x}_{m \times 1} + \mathbf{b}_{n \times 1}
    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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