How to find the equivalence classes on a relation?

Question

Suppose we define the relation $∼$ by $v∼w$ (where $v$ and $w$ are arbitrary elements in $R^n$) if there exists a matrix $$A∈ GL_n(R)$$ such that $v=Aw$. What are the equivalence classes for $∼$ in this case? NOTE:$GL_n(R)$ is a set that contains all the $n×n$ matrices with $det≠0$

Answer 1

The relation works like that: The zero vector is only in relation with itself. All the other vectors are in relation with each other

Proof: if $v=0$ then $Av=0$ for every matrix and therefore $0\sim w$ if and only if $w=0$.

Now take $v\not = 0$ and $w\not = 0$. I claim that $v\sim w$, since $v\not = 0$ you can find $v_1,...,v_{n-1}$ such that $\mathcal{B}_v:=\{v,v_1,...,v_{n-1}\}$ a basis, similarly find $w_1,...,w_{n-1}$ such that $\mathcal{B}_w:=\{w,w_1,...,w_{n-1}\}$ is another basis.

Now define a matrix by the following linear map : $$A(\lambda v + \lambda_1 v_1 +...+\lambda_{n-1} v_{n-1} ) := \lambda w+\lambda w_1 +...+\lambda w_{n-1}$$

(i.e. $A$ is the unique linear transformation that sends one basis to another this is usually denoted as $[I]^{\mathcal{B}_v}_{\mathcal{B}_w}$)

In particular you find a matrix $A$ which is invertible (because it sends a basis to itself), hence $\det A \not = 0$ such that $Av=w$. It follows that $v\sim w$.

Answer 2

Given any unit vectors, there is a nonsingular rotation matrix that takes the first vector to the other. Given any non-zero, non-unit vector, there is some non-singular scaling matrix that takes the vector to a unit vector. So given two arbitrary non-zero vectors $v$ and $w$, there are $S_v$ and $S_w$ such that $S_vv$ and $S_ww$ are unit vectors, and $R$ such that $R(S_vv)=S_ww$. Then $w=(S_w^{-1}RS_v)v$. So taking $A=S_w^{-1}RS_v$, we have that $v=Aw$, so given any non-zero vectors $v$, $w$, $v$~$w$; the set of non-zero vectors is an equivalence class.

Answer 3

Hint: There are two equivalence classes: $\{0\}$ and $\Bbb R^n\setminus \{0\}$.