Difference between the "functions" in calculus and the "functions" in Linear Transformations

Question

The word function in calculus refers to something like $f(x) = x^2+2x^3$ or $f(x) =\sin(x) $ etc....

In linear algebra, the word function is used like- A linear transformation is a function from $V \rightarrow W$.

And the functions of calculus like $f(x) = x^2+2x^3$ or $f(x) =\sin(x) $ etc. are actually vectors in either a polynomial space ($f(x) = x^2+2x^3$ ) or a Function space ( like $f(x) =\sin(x) $ ) .

Now the word function in Linear Algebra is used twice as I showed above.

So according to me the functions of calculus are just vectors in linear algebra. Is this correct or not.

But then what are the functions being used in the definition of Linear Transformations. And how are they different from the functions of calculus and the functions which are vectors in linear algebra.

Edit:

Why is the graph of a linear transformation from any vector space to any other vector space not always a straight line. Can anyone give any counter examples.

Answer 1

The short answer is: Context matters!

The word "function" appears in many (if not all) different branches of mathematics, where the quality they have in common is that a function $f\colon X\to Y$ is a mapping between sets.

In Calculus, we often think of functions as mappings from a subset of $\mathbb{R}$ to $\mathbb{R}$ which satisfy some regularity condition (continuous, differentiable, analytic, measurable, integrable...), and sometimes we assume implicitly that the function we are talking about has those desired properties.

In Linear Algebra, the "functions" we consider are linear maps from a vector space $V$ to another vector space $W$. So in many instances, if some statement starts with "Let $f\colon V\to W$ be a function", it usually means a linear mapping.

In Topology, a function $f\colon X\to Y$ usually means a continuous mapping between two spaces.

As for what you said: yes, it's true that the functions $f\colon A\subseteq \mathbb{R}\to \mathbb{R}$ are abstract vectors of some space!

So, to summarize: a function is a mapping between sets, but depending on the context, that mapping can be required to have some additional properties.

As a side note, some people like to reserve the concept "function" for mappings with codomain $\mathbb{R}$ (or a field in general) and call everything else "map". So a linear transformation $f\colon \mathbb{R}^{2}\to \mathbb{R}$ is refered to as a function, and a linear transformation $f\colon \mathbb{R}^{2}\to \mathbb{R}^{2}$ may be called just a map.

Edit: say you have $y=ax+b$, where $a$ and $b$ are real numbers. That equation defines a map $f\colon \mathbb{R}\to \mathbb{R}$ given by $f(x)=ax+b$. This map is a "function" in the sense of calculus (and it has practically every property you would like). It is also a map between vector spaces, but it may not be linear (if $b\neq 0$ it isn't), so it would not be considered an "interesting function" between vector spaces (it is an affine map, to be exact).

Still, it is a vector of many vector spaces: for example, it is in the following spaces:

$$V=\{\text{Polynomials in one variable}\}$$ $$W=\{\text{Mappings from } \mathbb{R} \text{ to itself}\}$$ $$F=\{\text{Affine maps from }\mathbb{R}\text{ to itself}\}$$

Answer 2

Generally function $f=(F,A,B)$ is defined by triple, where $A$, $B$ are sets, $F$ is functional graph and domain $pr_1F=A$.

Answer 3

A function is defined as a relation between two sets that maps one element from one set to exactly one of the other set. For your example $f(x) = x^{2} + 2x^{3}$, the element $x$ in the domain is mapped to the codomain element $x^{2} + 2x^{3}$.

In your linear algebra example, your domain is denoted $V$ and your codomain is denoted $W$.

A linear transformation is one specific type of function where an extra restriction is required: $f(cx + y) = cf(x) + f(y)$. Both are examples of functions but this restriction placed on linear maps may or may not hold for functions in general.

Linear transformations can be graphed but they are commonly graphed as vector fields; a linear transformation graph would not look like your typical one-to-one function from calculus.