# Is R^n Euclidean Space a vector space too?

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1. Sep 4, 2015

### bacte2013

Dear Physics Forum personnel,

I am curious if the euclidean space R^n is an example of vector space. Also can matrices with 1x2 or 2x1 dimension be a vector for the R^n?

Sincerely,

PK

2. Sep 4, 2015

### micromass

Staff Emeritus
Yes.

No clue what you mean.

3. Sep 4, 2015

### bacte2013

As for the second question, I mean if the 1x2 matrix (a1, a2) or its 2x1 form (column vector) can be considered as a vector for the R^2 since the R^2 is basically the collection of real numbers in the ordered pair (a, b)?

4. Sep 4, 2015

### micromass

Staff Emeritus
There is a linear isomorphism between the $1\times 2$-matrices and $\mathbb{R}^2$, yes.

5. Sep 4, 2015

### HallsofIvy

Staff Emeritus
The Eulclidean space Rn is geometric- there are such things as points and distances defined but no "operations". A vector space is algebraic we must have operations such as sum and scalar multiplication defined. Of course, for, finite dimensional Rn, we can define the sum as (x1, x2,... , xn)+ (y1, y2, ..., yn)= (x1+ y1, x2+ y2... , xn+ yn) and scalar multiplication defined as a(x1, x2, ..., xn)= (ax1, ax2, ..., axn). If we consider those operations as "natural" then we can think of this as a "natural" correspondence between Rn and an n dimensional vector space.[/sub][/sub]