Finding an isomorphism between spans

Question

The qustion is how to find an isomorphism between two spans: span{(1,0,0,0),(0,1,0,0)} and span{(3,2,1,0),(1,1,1,1)}. I can see they are both lineary independet so they form bases, but how do I find the isomorphism?

Answer 1

A linear mapping between the spans (subspaces) can be given by mapping a basis of span to the basis (generating system in general) of the other span, here

$\phi(1,0,0,0) = (3,2,0,0)$ and $\phi(0,1,0,0)=(1,1,1,1).$

Since both spans have the same dimension you already have an isomorphism.

Answer 2

Hint: Any two finite dimensional vector spaces of the same dimension are isomorphic. Let $\{v_1,\dots,v_n\}$ and $\{w_1,\dots,w_n\}$ be bases. You can map $v_i\mapsto w_i$, and extend linearly. This defines an isomorphism.