# What is a nonconstant linear function?

1. ### spoke

5
arent linear functions always constant?

2. ### espen180

836
No, they aren't. Concider for example f(x)=x.

3. ### Bacle2

1,179
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.

4. ### spoke

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.

5. ### Office_Shredder

4,500
Staff Emeritus
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)

6. ### Bacle2

1,179
spoke:

You may be confusing constant rate of change, i.e., constant derivative--a property of linear functions-- with constant function.

7. ### spoke

5
So would this relation be an example constant function? {(1,2), (2,2), (3,2), (4,2)}

8. ### Bacle2

1,179
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).

9. ### Dickfore

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.

10. ### Bacle2

1,179
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.

11. ### Dickfore

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 $\mathrm{rank}A \le m < n$.