Your polynomial, f(x) = a + bx +cx^2 + dx^3, is an element of the vector space P3®, the polynomial vector space of degree at most 3 over the reals. This space is isomorphic to R^4 and it has a standard basis: {1, x, x^2, x^3}. Then you can see that any such f(x) may be written as a linear combination of the basis vectors with real valued scalars.
As an exercise, you can check that P3® satisfies some of the properties of vector spaces yourself (existence of zero vector, associativity and commutativity of vector addition, distributivity of scalar multiplication over vector sums).
What happens to elements with powers of x above 3? Say we multiply the example vector above with itself. We would end up with a component d2x6, witch is not part of the P3R vector space, right?
Do we need a special multiplication rule to handle powers of x above 3?
I’ve worked with quaternions before, which has " special" multiplication rules by defining i j and k.
Multiplication of two vectors is not an operation defined on vector spaces. If you want that, you’re looking at either a structure known as an inner product space or an algebra over a field.
Note that the usual notion of polynomial multiplication doesn’t apply to polynomial vector spaces, nor does it agree with the definition of an inner product nor the bilinear product of an algebra.
That’s only if you’re working with the perspective of it being a polynomial. When you’re considering the polynomial as a vector however, that operation simply doesn’t exist
Your polynomial, f(x) = a + bx +cx^2 + dx^3, is an element of the vector space P3®, the polynomial vector space of degree at most 3 over the reals. This space is isomorphic to R^4 and it has a standard basis: {1, x, x^2, x^3}. Then you can see that any such f(x) may be written as a linear combination of the basis vectors with real valued scalars.
As an exercise, you can check that P3® satisfies some of the properties of vector spaces yourself (existence of zero vector, associativity and commutativity of vector addition, distributivity of scalar multiplication over vector sums).
What happens to elements with powers of x above 3? Say we multiply the example vector above with itself. We would end up with a component d2x6, witch is not part of the P3R vector space, right?
Do we need a special multiplication rule to handle powers of x above 3? I’ve worked with quaternions before, which has " special" multiplication rules by defining i j and k.
Multiplication of two vectors is not an operation defined on vector spaces. If you want that, you’re looking at either a structure known as an inner product space or an algebra over a field.
Note that the usual notion of polynomial multiplication doesn’t apply to polynomial vector spaces, nor does it agree with the definition of an inner product nor the bilinear product of an algebra.
That’s only if you’re working with the perspective of it being a polynomial. When you’re considering the polynomial as a vector however, that operation simply doesn’t exist