# All linear functions are homogeneous of degree one?

I was looking through the Wikipedia page of "Homogeneous functions" and it stated that any linear function that maps V onto W is homogeneous of degree one. However, when I try to apply the definition of a homogeneous function to a line defined by one variable "v" and a non-zero constant "z":

f(v)=v+z

f(tv)=(tv)+z

f(tv)=t(v+z/t)≠tf(v)

I find that the line is not homogeneous.

So why is it that a linear function is always homogeneous even though the example I gave shows that it's not? What conceptual/calculation mistake did I make in in the process?

• By the definition in (en.wikipedia.org/wiki/…) though your function wouldn't be a linear map anymore. The page on (en.wikipedia.org/wiki/Linear_function) defines "linear function" either as a polynomial of degree 0 or 1 (that is, $y=ax+b$ or $y=c$) or a linear map. For the above problem, the linear maps must have been meant. Article should be updated for clarity though. – Maximilian Gerhardt Jun 21 '16 at 23:31
• See the definition of "linear function" that applies to vector spaces. I assume $V,W$ are vector spaces. – GEdgar Jun 22 '16 at 0:25

a homogeneous function is a polynomial function which all the terms have the same degree. then in your example (in this case of one dimension) $v \rightarrow av + z$ is not a homogeneous polynomial since $z$ is a vector constant, not a variable.