What is the difference between ColA coinciding with R^3 and spanning a subspace?

[fk378]

What exactly does it mean when ColA "coincides with" some R^n, say, R^3? What is the difference between coinciding with and spanning the subspace?

 

[Pere Callahan]

What du you mean by ColA?

If a space coincides with R^n then in my understanding it is R^n although for finite-dimensional vector spaces it might be confusing that every real vector space of dimension n is isomorphic to R^n and can therefore be said to coincide with R^n.

Another point to consider is that if you are working in some higher dimensional R^n, say R^5, than there a bunch of R^3's canonically embedded into that R^5. namely those which are the span of three basis vectors, and the wording "to coincide with R^3" might mean "to be one of those R^3's being the span of three basis vectors".

the differnce bettween coincide with some subspace and spanning this subspace is that only two subspaces can coincide while some (finite) set of vectors can span this subspace. A subspace and a set of vectors spanning the subspace are closely related, yet different things.

 

[HallsofIvy]

The "column space" of matrix A? If the columns of A (thought of as vectors) span a certain space then the column space of A is that space.