Is R^n Euclidean Space a vector space too?

[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?

PK

 

[micromass]

I am curious if the euclidean space R^n is an example of vector space.

Yes.

Also can matrices with 1x2 or 2x1 dimension be a vector for the R^n?

No clue what you mean.

 

[bacte2013]

Yes.
No clue what you mean.

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)?

 

[micromass]

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

 

[HallsofIvy]

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,... , x_n)+ (y₁, y₂, ..., y_n)= (x₁+ y₁, x₂+ y₂... , x_n+ y_n) and scalar multiplication defined as a(x₁, x₂, ..., x_n)= (ax₁, ax₂, ..., ax_n). 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]