Picking a specific basis is often looked upon with disdain when making statements that are about basis independent quantities. For example, one might define the trace of a matrix to be the sum of the diagonal elements, but many mathematicians would never consider such a definition since it presupposes a choice of basis. For someone working on algorithms, however, this might be a very natural perspective.

What are the advantages and disadvantages to choosing a specific basis? Are there any situations where the "right" proof requires choosing a basis? (I mean a proof with the most clarity and insight -- this is subjective, of course.) What about the opposite situation, where the right proof never picks a basis? Or is it the case that one can very generally argue that any proof done in one manner can be easily translated to the other setting? Are there examples of proofs where the only known proof relies on choosing a basis?