# Can you say whether some set is the vector space without explicitly defining vector addition and scalar multiplication?

Can you think of any examples of vector spaces?

He proceeds by saying:

The Euclidean spaces of the last chapter – V = $$\mathbb{R^{2}}, \mathbb{R^{3}},..., \mathbb{R^{n}}$$ – are all examples of vector spaces.

Well, how can you say whether some set is a vector space without explicitly defining vector addition and scalar multiplication?

To make my question more concrete, let's consider an example, say $$\mathbb R^{2}$$. We have set, call it $$V$$, that consists of the vectors $$(x,y)^{T}$$

Consider two arbitrary vectors , $$\mathbf{v} = (a,b)^{T}$$, $$\mathbf{u} = (c,d)^{T}$$ such that both vectors are in $$V$$

$$\begin {pmatrix} a_{1} \\ a_{2} \\ \end {pmatrix} + \begin {pmatrix} b_{1} \\ b_{2} \\ \end {pmatrix} = \begin {pmatrix} a_{1} -b_{1} \\ a_{2} -b_{2} \\ \end {pmatrix}$$

So we have $$\mathbf{v+u} = \begin {pmatrix} a \\ b \\ \end {pmatrix} + \begin {pmatrix} c \\ d \\ \end {pmatrix} = \begin {pmatrix} a-c \\ b-d \\ \end {pmatrix}$$

And $$\mathbf{u+v} = \begin {pmatrix} c \\ d \\ \end {pmatrix} + \begin {pmatrix} a \\ b \\ \end {pmatrix} = \begin {pmatrix} c- a \\ d-b \\ \end {pmatrix}$$

Provided that $$\bf u ≠ v$$, we have

$$\mathbf{u+v} ≠ \mathbf{v+u}$$

And hence, in our case, one of the axioms fails, implying that $$V$$ is not a vector space.

So coming back to my question, is it correct to say that $$\mathbb R^{n}$$ forms a vector space without explicitly stating what you mean by vector addition and scalar multiplication?