Invertible transformations

1. Jun 10, 2013

Bipolarity

For a linear transformation to be invertible, is it a requirement that the domain and codomain be the same vector space, or merely that they have the same dimension? My intuition tells me they merely need the same dimension but someone can correct me please?

BiP

2. Jun 10, 2013

Office_Shredder

Staff Emeritus
You have to be a bit careful - the range of the linear transformation needs to have the same dimension, not just the codomain. But there is no requirement they be exactly the same vector space (in fact two vector spaces which are the same dimension are always isomorphic, so in a sense they are the same vector space as far as any linear algebra operations go)

3. Jun 10, 2013

mathwonk

your intuition is correct. two vector spaces are isomorphic precisely when they have the same dimension.

4. Jun 11, 2013

HallsofIvy

Staff Emeritus
For example, the function f(a, b)= a+ bx is an invertible linear transformation for R2, the vector space of ordered pairs of real numbers, to P2, the vector space of linear polynomials, both of which have dimension 2.