Are vectors and (not necessarily linear) transformations/maps the same thing?

Question

As I learn more mathematics, I increasingly notice similarities between concepts that are otherwise identified differently. I only have knowledge of basic linear algebra, but it seems to me that a vector and a transformation/map are the same thing?

If I have the vector $\overrightarrow{r}(x, y) = (xy, x+2, 3)$, then it is mapping from $\mathbb{R}^2 \to \mathbb{R}^3$, right?

The parameterisation $\overrightarrow{r}(t) = (t, t^2)$ is mapping from $\mathbb{R} \to \mathbb{R^2}$.

I would greatly appreciate it if people could please take the time to clarify this.

P.S. I'm not entirely sure what to tag this with, but I suspect it falls under the category of abstract algebra. If not, then I would appreciate it if people could kindly edit my question with the appropriate tags.

Answer 1

Transformation and map are functions. So in a general sense they are the same. Vectors are not functions.

One may call $\overrightarrow r$ a vector field, which is an old term. In general, a vector field is a map $\mathbb{R}^{n} \to \mathbb{R}^{m}$. (see Apostol's Calculus II, for instance).

Every element of a vector space is called a vector. For example, a point of $\mathbb{R}^{n}$, a vector space, is called a vector for every integer $n \geq 1$.

Answer 2

i am giving you certain example that is linear transformation consider $$f:R\to R$$ $$f(x)=exp(x)$$ you see that this is linear transformation but there is no vector involve here so you can conclude that vectors and transformations are not same.....