# Finding a Matrix to Connect Equivalence Classes in Quotient Space

• mr.tea
In summary, the conversation is about showing that the quotient space ##\mathbb{R}\small/ \sim## has two elements, with the first equivalence class being ##\{0\}## and the second being ##\mathbb{R}^n - \{ 0 \} ##. The discussion also involves finding a matrix that connects two given vectors in ##\mathbb{R}^n##, which can be done by creating a basis with both vectors and using a matrix that swaps them and leaves the rest invariant.
mr.tea

## Homework Statement

(part of a bigger question)
For ##x,y \in \mathbb{R}^n##, write ##x \sim y \iff## there exists ##M \in GL(n,\mathbb{R})## such that ##x=My##.
Show that the quotient space ##\mathbb{R}\small/ \sim## consists of two elements.

## The Attempt at a Solution

Well, it is easy to see that the first equivalence class is ##\{0\}##. This means that the second equivalence class should be ##\mathbb{R}^n - \{ 0 \} ##.
But I don't know how to show that given ##x,y\in \mathbb{R}^n##, I can find a matrix that connects between them.
Any suggestions will be great.

Thank you!

If they are linearly dependent, it's easy. If not, why not make a basis with both of them?

fresh_42 said:
If they are linearly dependent, it's easy. If not, why not make a basis with both of them?
However, I still don't get how it will help me with taking them both and complete it to a basis.

The matrix (expressed in the coordinates of the new basis) that simply swaps the two and leaves the rest invariant shouldn't be that hard to figure out.

## 1. What is a quotient space in topology?

A quotient space in topology is a mathematical concept used to describe the relationship between two spaces that are related by an equivalence relation. In simpler terms, it is a space that is obtained by "gluing" together points of another space according to certain rules or identifications.

## 2. How is a quotient space defined?

A quotient space is defined as the set of all equivalence classes of the original space under the equivalence relation. This means that each point in the quotient space represents a collection of points in the original space that are considered equivalent.

## 3. What are some examples of quotient spaces?

Some examples of quotient spaces include the quotient of a rectangle by identifying opposite sides, the quotient of a sphere by identifying antipodal points, and the quotient of a torus by identifying opposite sides of a square.

## 4. What is the importance of quotient spaces in topology?

Quotient spaces play a crucial role in topology as they allow for the study of topological spaces with more complicated structures by simplifying them into more manageable spaces. They also help to identify topological properties that are preserved under continuous maps.

## 5. How are quotient spaces related to topological equivalence?

Quotient spaces are closely related to topological equivalence, as two spaces are considered topologically equivalent if there exists a homeomorphism between them. In other words, if the two spaces can be transformed into each other by stretching, bending, or shrinking without tearing or gluing, they are topologically equivalent.

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