Example of a vector space with different addition/scalar multiplication operators

Question

I have learned that a vector space is a set of elements called vectors, on which are defined an addition operation and a scalar multiplication operation with the scalars in some field $F$. However, all the examples of vector fields I know ($\mathbb{R}^n$, $\mathbb{C}^n$, $\mathbb{F}^{(n, n)}$, etc) all use the "normal" operations of addition and scalar multiplication.

As an example, let's consider the set $S = \{(x, y) \in \mathbb{R}^2 \lvert x + y = 10\}$, for instance. With the intuitive definitions of addition and scalar multiplication ($(a, b) + (c, d) = (a + c, b + d)$ and $k(a, b) = (ka, kb)$), I can verify that this is a vector space, but I can't think of any other definitions of the addition/scalar multiplication operation that still makes this set a vector space.

Could someone provide and explain a nonobvious example of addition/scalar multiplication operations that still keeps the set as a vector space?

Answer 1

This is not an obvious example, but you can prove it by straightforward computations.

Consider $V=\{(x,y,z)\in \mathbb{R}^3: x>0\}$ with the following operations

$$ (x,y,z)\oplus (a,b,c)=(x\cdot a,y+b+3,z+c)$$ $$ \alpha \otimes (x,y,z)=(x^\alpha,\alpha \cdot y +3(\alpha -1),\alpha \cdot z)$$

Of course, $+$ and $\cdot$ are the usual sum and multiplication. It can seem weird, but the null element is $(1,-3,0)$.

Answer 2

An example that I remember from my first year at university:

Take $V=(0,\infty)$ with $v_1\oplus v_2 = v_1 v_2$ and $k\otimes v = v^k$ for $v,v_1,v_2\in V$ and $k\in\mathbb R.$

This is though isomorphic with $\mathbb R$ by the isomorphism $\phi(v) = \ln v.$