Invertible transformations

[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

 

[Office_Shredder]

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)

 

[mathwonk]

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

 

[HallsofIvy]

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.

 

[blue_raver22]

olarBear, you are correct in your intuition. For a linear transformation to be invertible, it is not necessary for the domain and codomain to be the same vector space. However, it is necessary for them to have the same dimension. This is because the invertibility of a linear transformation is determined by the existence of a unique inverse matrix, which is only possible when the dimensions of the domain and codomain match. In short, as long as the dimensions match, the linear transformation can be invertible, regardless of whether the domain and codomain are the same vector space.