# intutive difference between linear map/transformation vs linear function

my linear algebr textbook defines a linear transformation/map as one that satisfies:

i. T(u+v)=T(u) +T(v). ii. T(cu) = cT(u)

However, what is traditionally called a linear function, in non-abstract algebra (or highschool algebra, or whatever it is formally called), namely: f(x) = a + bx is not a linear mapping according to the linear algebra definition, unless a = 0.

Is there an intuitive reason why the first definition is called a linear map, and why you would not call y=1+x a linear map, despite the fact that it defines a straight line on a plane? I could simply take the definitions as given, but it always helps me to have an intuitive understanding of such terminology.

• Some curricula do not make the mistake you clearly describe and call affine function every real function $x\mapsto ax+b$ and linear function only the real functions $x\mapsto ax$. This way, the one dimensional case is in line with the higher dimensional case, where linear is equivalent to being $X\mapsto AX$ for some matrix $A$.
– Did
Sep 3, 2016 at 11:17
• The convention in high school is inconsistent with the one in more advanced mathematics, and you have to keep track of that. In particular, you should keep this in mind when you talk to or teach someone who has never learned any math beyond high school. It's worth noting that since there is no ultimate authority in terminology and notation, such inconsistencies are everywhere. You have to rely on explicit definitions or context in order to figure out the exact definition of something in something you are reading. Jan 30 at 15:16

the reason is that a linear function does not preserve the origin. but a linear map with the properties you listed does!

example of linear function: $$f(x)=a x +b$$ \begin{align} f(u+v)=a(u+v) + b =au + av + b = f(u)+f(v)-b \neq f(u) + f(v) \end{align}

example of linear map: $$g(x)=A x$$ \begin{align} g(u+v)=A(u+v)=Au+Av=g(u)+g(v) \end{align}

With a linear function you cannot transform a vector space into another vector space, thing that you can do with a linear map.

So now comes the intuitive way of seeing it: A linear map takes vectors and rotates and scales them and project them onto a subspace (not necessarily). A linear function does the same plus in the end it translates the origin, applying a translation distrupts many beatiful and USEFUL properties.

Remark: In general $$x$$ is column vector with N elements and $$a,b,A$$ matrices with K rows and N columns. But this example works in the one-dimensional case too ($$K=N=1$$)

• In other words, calling linear the function $x\mapsto ax+b$ on the real line, when $b\ne0$, is contra-intuitive...
– Did
Sep 3, 2016 at 11:18
• i agree almost entirely with you. Sep 3, 2016 at 12:27