What is the difference between linear function and linear map(transformation)?

Question

I know that linear mapping is

    `T: V-> W where V,W are vector spaces` 

and linear function is

f: V->F where V is a vector space and F is a field

But I don't really know the difference intuitively. What are the examples of linear map and linear function? And is linear mapping included in linear function?

Answer 1

A linear function (or functional) gives you a scalar value from some field $\mathbb{F}$. On the other hand a linear map (or transformation or operator) gives you another vector. So a linear functional is a special case of a linear map which gives you a vector with only one entry.

Answer 2

There is less in common than the name suggests.

A linear function $y = mx + b$ is a function between relating two variables such that their graph is a line.

Actually a linear function can map from a vector space to a field (e.g. the real numbers)

$f(x_1, x_2, x_3,\cdots, x_n) = a_1 x_1 + a_2 x_2 +\cdots + a_nx_n + b$

A linear transformation (linear map) is a function between vector spaces such that $T(\mathbf u+\mathbf v) = T(\mathbf u) + T(\mathbf v)$

A linear function is a linear map if and only if $b= 0.$

Historically, the study of linear transformations (Linear Algebra) began with studies of systems of linear equations, and that is where the name comes from. But, it is traveled so far from this starting point, that it has become an entirely different animal. At this point the name is a little bit misleading.