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I'm reading book on linear algebra where author asks

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 $V$

And define vector addition as:

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

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?

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No, it is not correct. But the author of the textbook that you mention is talking about “The Euclidean spaces of the last chapter”. It is very likely that these spaces are introduced there not simply as sets but as sets endowed with a sum and a product by a scalar.

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Euclidean space has a standard definition of vector addition and scalar multiplication. But it's possible to come up with different vector spaces by changing those definitions. You can find examples in your text. When you define a different addition and scalar multiplication, you then need to check if the vector space axioms are satisfied, to determine if it's a vector space or not.

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