I'm taking a course in Linear Algebra right now, and am having a hard time wrapping my head around bases, especially since my prof didn't really explain them fully. I would really appreciate any insight you could give me as to what bases are! Also, can there can be multiple different bases for a single subspace?
Thanks in advance.
On
While a bit late to the game, I thought another perspective might help.
Consider the following physical example. Now, without being too pedantic about definition, a basis for a vector space is much like a building block of a biological system. We can build a human body from a set of cells. That is, we can construct all aspects of our anatomy beginning with a certain set of cells (e.g. nerve cells, blood cells, germ cells, epithelial cells, etc). Thus, if we take our various tissue as vectors, then we have as a basis our cells.
But we could certainly have another biological basis from which to build our biological vectors. Namely, biomolecules. Indeed, we could express our other defined basis using this basis. Thus, our biological vector space has more than one biological basis.
Some might argue that there's not a "full correspondence" here with the mathematical notion of a basis for a vector space because, for example, how could one exhibit a change of basis from biomolecules into cells (i.e. how does one express a biomolecule as "linear combination" of cells)? But I argue the idea of building blocks captures the underlying spirit of a basis for a first pass.
They are subsets that “efficiently capture” the rest of the vector space. A sort of skeleton, if you will, or maybe like compressing a computer file.
This means that you can recover every other element in the space by using just the operations (scalar multiplication and addition) and furthermore there were a exactly one way (in a sense) to generate each element.
Finally, linear transformations (a main object of study) completely determined by what they do to a basis. You can see how this makes finite dimensional vector spaces things easier if you can forget about the potentially infinite number of vectors and just focus on what a finite subset does, and trust the other elements to follow suit.
It can happen that a subspace has infinitely many distinct sets which are bases. Even for a $1$ dimensional space over an infinite field, there are infinitely many.