Question

Quotient spaces.

Answer 1

linear algebra?//

Answer 2

yeah, just give me a sec need to type LaTeX stuff

Answer 3

Let V be a K-vectorspace = R^n and W a subspace of V. W is isomorphic with R^m
Let \[ f: V \rightarrow V/W : v \mapsto v+W\]
be a linear function.

I am confused by the quotientspace V/W. According to the book it's a vectorspace with all affine subspaces of V. But if you translate all W's, you get the vectorspace V again, right?

That's why this gets me confused: V/W is isomorphic with R^(n-m).
Is this because all elements of V/W are actually just a representative for an equivalence class of V/W?

Answer 4

sorry cant help you with it.......
i have studied quotient groups......
but this topic we have in next sem

Answer 5

as much as I know, quotient space is a space defind from other space by using some equivalence relation.

Answer 6

in this way you can get a sphere from a square , for example

Answer 7

I'm just confused that it's isomorphic with R^(n-m). because it essentially geometrically spans R^n. The only plausible explanation I can think of is that the elements of V/W are actually just representatives for equivalence relations.

Answer 8

equivalence classes*

Answer 9

a quotient space will be all vectors that have norm(v)=0, where norm is "length", :). So, all vector is equivalent to it self, and it's norm therefore is 0, so it will be in this quotient space. But also there will be vectors that are obtained from vectors of W by mooving them by v.