# Polynomials are vectors Why is it that polynomials are vectors? Not simply *similar to vectors*, but *truly vectors* (at least in the context of linear algebra). The reason is that: * They can be added together * They can be multiplied by numbers * The remaining properties required for a vector space (*commutativity, additivity, associativity*) are satisfied We can note that when considering all possible polynomials, they clearly form a vector space. This is because: * If we add two polynomials together we get another polynomial * If we multiply a polynomial times a number, we get another polynomial We can also note (again, this will be *inspired by geometry*, as are most points in linear algebra) that just as when two geometric vectors are constrained to a plane and we can never escape that plane via a linear combination of the two vectors (i.e. the two vectors are stuck in a subspace), can this occur with polynomials? Can polynomials be stuck in a subspace? First we must note: polynomials are *not* geometric vectors. You truly could not find two vector spaces that are any more different than geometric vector spaces (drawings) and polynomials (abstract symbols). Perhaps the only thing they have in common is their ability to be added together and be multiplied by numbers. With that said, we can indeed find an analogue of being *stuck in some subspace* for polynomials, even though there is *nothing geometric about it*! It was indeed *inspired* by geometry, but it is *not geometric*. ### The most fundamental source of power of linear algebra There is nothing geometric about polynomials. However, they **display a phenomenon** that we cannot help but associate with our geometric intuition for geometric vectors. This happens throughout linear algebra and is considered by some to be the most fundamental source of the power of linear algebra. --- Date: 20220607 Links to: Tags: #review References: * []()