This question might sound stupid to some of you, but i'm in the process of figuring out this whole coordinates subject.

Lets say V is an n dimensional vector space.

and lets say A and B are two different bases of V.

let v be vector in V.

now let $\displaystyle (\alpha_1, \alpha_2...\alpha_n)$ be the coordinates vector of v with respect to A,

and let $\displaystyle (\beta_1, \beta_2...\beta_n)$ be the coordinates vector of v with respect to B.

now, my question is this:

geometrically speaking; are $\displaystyle (\alpha_1, \alpha_2...\alpha_n)$ and $\displaystyle (\beta_1, \beta_2...\beta_n)$ the same vector? (i know that if n>3 then there's no geometric meaning to vectors, so let's say we're talking about 2 or 3 dimensional space)

and another one:

let's take for example $\displaystyle b_i\in B$.

what are its coordinates with respect to B?

thanks in advanced.