# What is the difference between linear mappings and linear functions? [duplicate]

Let $V$ and $V'$ be vector spaces over a field $K$. A linear mapping $$f:V \to V'$$ is a mapping which preserves addition and scalar multiplication.

My question is: what is the difference between linear mappings and linear functions?

• None. ${}{}{}{}$ Commented Mar 30, 2016 at 16:53
• I'd suppose a linear function takes values in $\mathbf R$ – in which case it would be better known as linear form. Commented Mar 30, 2016 at 16:58
• When I was in school, a linear function was just a line, i.e., $f(x) = ax+b$. In general, a function usually maps to $\mathbb R$ or $\mathbb C$, whereas a map or mapping can map to anything. Commented Mar 30, 2016 at 17:01
• For some people, none. But for example $ax+by+c$ is traditionally called a linear function, though if $c\ne 0$ it should more precisely be called affine. Commented Mar 30, 2016 at 17:12
• Relevant, almost dup: math.stackexchange.com/questions/95741/… Commented Mar 30, 2016 at 17:57

Depends on the definition you are using. Linear mappings are typically the same as linear functions, although in some contexts, a linear function is strictly some function in the form of $y = mx+b$.

Linear mappings are functions that map two domains and whose operations have linearity, that is the linearity of scalar multiplication and addition.

For the definition of linearity it still the same for functions or mappings.

But the term mapping has more general uses than than function, a mapping could be a function, functional or operator...

However, these three terms are not the same. A function is a term you used for mappings that map a set A to B where A and B are subsets of real of complex numbers or almost any map into $R^n$ or $C^n$ .
A functional is a mapping that is defined from a subset of a space X into the space of real or complex numbers. Finally an operator is defined from a space X to a space Y, where X, Y could sequence or function spaces.

Here an examples that clearfy more the difference between these three terms, let the function $f:R \rightarrow R$ such that $f(x)= x$.

Let us integral this function between $0$ and $1$, then $\int^{1}_{0}x =\frac{1}{2}$.

You can notice that the defined integral is a functional, it mapped a function ( x ) into a real ( 1/2).

However the undefined integral of this function is a function, $\int x =\frac{1}{2}x^2$.

Thus, the undefined integral is an operator.

You can use the term of mapping for all previous examples but the term function can be uses only for the first case.

• How are those two last not functions? Commented Mar 30, 2016 at 18:05
• Roughly speaking functions map points to points, Functionals map vectors to points and operators maps vectors to vectors Commented Mar 30, 2016 at 18:07
• None of those are meaningful distinctions. Commented Mar 30, 2016 at 18:10
• If you go with the somewhat naive definition that a function is something that takes an input and gives an output, functionals and operators do exactly that. If you take the better definition that a function is a certain special kind of relation, then functionals and operators are exactly that. I'm afraid that this answer is nonsense.
– user296602
Commented Mar 30, 2016 at 18:33
• Not just an input but a point, that has no directions and no movements however the others do that is what I meant when I used the term vector A function maps a real to a real (or complex), functional maps a set from function space or sequence space to a real (or complex). However, the operators is from a sequence or fonction space to sequence or fonction space Commented Mar 30, 2016 at 18:36