Our definition of spanning a vector space is that every vector in the space is a linear combination of vectors from the set.

Currently, our definition of linear independence is that no vector in the set is a linear combination of other vectors from the set, which we recharacterize as the zero vector is a unique linear combination of vectors from the set.

What does the community think about changing our definition to if a vector in the space is a linear combination of vectors from the set, then that linear combination is unique?

I like this, because we immediately get that a spanning and independent set satisfies every vector in the space is a unique linear combination of vectors from the set from the definitions.

It also means our canonical vector equation characterizations line up a bit better: both involve the same vector equation, but for spanning we say "has at least one solution" and for independent we say "has at most one solution".

Thoughts @siwelwerd @jford1906 @jkostiuk and others?