As I study linear algebra, and geometric "arrows" and polynomials composed of variables, are both treated as "vectors," I've begun to wonder about the conceptual link between them.
Both polynomials and vectors can be drawn on a piece of graph paper with a Cartesian grid for a basis.
I know all the rules of linear combinations, etc, and yes, I understand that polynomials and geometric vectors are distinct objects, but can somebody please provide a visual, intuitive explanation to the link I see in front of me on the graph paper? Like, "the geometric vectors point to where the two polynomials intersect..." or something (just an example; I know that is wrong).
Is the visual connection I see a complete illusion? Or is there some nifty connection there? Just curious, thanks again!
Think of it this way, with the basis $e_1,e_2,e_3$ you can create three dimensional objects as well as lines and points. By contrast, with the basis $1, x, x^2$ you can create families of quadratics, lines, and also points. So starting with the first element of the basis, namely $e_1$, ask yourself what you can "visually" compose. Then, start with the first element of the next basis, namely: {1}. Successively add on elements to each basis and it will paint somewhat of a picture. On a side note, I think it may be a stretch to create a solid visual relationship between the two vector spaces. This mental sketch is a worthwhile exercise for you question.