1. ## General question about change of linear basis

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 $(\alpha_1, \alpha_2...\alpha_n)$ be the coordinates vector of v with respect to A,
and let $(\beta_1, \beta_2...\beta_n)$ be the coordinates vector of v with respect to B.

now, my question is this:
geometrically speaking; are $(\alpha_1, \alpha_2...\alpha_n)$ and $(\beta_1, \beta_2...\beta_n)$ 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 $b_i\in B$.
what are its coordinates with respect to B?

2. ## Re: General question about change of linear basis

Take the vector v = $\begin{bmatrix}1 \\ 1 \end{bmatrix}$
A = $\begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix}$ is a basis
B = $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is a basis
the coordinates for v with respect to basis A is $\begin{bmatrix}-3 \\ 2 \end{bmatrix}$
the coordintes for v with respect to basis B is just $\begin{bmatrix}1 \\ 1 \end{bmatrix}$
By geometrically the same vector, perhaps you means do both of these coordinates are of the same magnitude and direction.
Right of the bat we see they are not of the same magnitude so regarless of direction they are not the same vector.

3. ## Re: General question about change of linear basis

Hi jakncoke.

obviously $\left [ v \right ]_{A}$ and $\left [ v \right ]_{B}$ are not the same, so perhaps i should've asked: do $\begin{bmatrix}1 \\ 1 \end{bmatrix}$ with respect to A, and $\begin{bmatrix}1 \\ 1 \end{bmatrix}$ with respect to B - both point to the same direction?

thank you!

4. ## Re: General question about change of linear basis

if they point in the same direction then one vector would be a multiple of the other which is not the case.