# what is a nonconstant linear function?

by spoke
Tags: function, linear, nonconstant
 P: 5 arent linear functions always constant?
 P: 836 No, they aren't. Concider for example f(x)=x.
 Sci Advisor P: 1,168 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.
P: 5

## what is a nonconstant linear function?

 Quote by espen180 No, they aren't. Concider for example f(x)=x.
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.
 Mentor P: 4,499 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)
 Sci Advisor P: 1,168 spoke: You may be confusing constant rate of change, i.e., constant derivative--a property of linear functions-- with constant function.
P: 5
 Quote by Office_Shredder 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)
So would this relation be an example constant function? {(1,2), (2,2), (3,2), (4,2)}
 Sci Advisor P: 1,168 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).
 P: 3,015 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.
 Sci Advisor P: 1,168 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.
P: 3,015
 Quote by Bacle2 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.
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 $\mathbb{C}^m \rightarrow \mathbb{C}^n$. But, now, the function may be constant in a more general case, when $\mathrm{rank}A \le m < n$.

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