Comment on Explain yourselves, comp sci.
chonglibloodsport@lemmy.world 2 months agoYour 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 queen written as a linear combination of the basis vectors with real valued scalars.
i_love_FFT@lemmy.ml [bot] 2 months ago
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 d^2x^6, witch is not part of the P3R vector source, 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.
chonglibloodsport@lemmy.world 2 months ago
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. For that you need an algebra.
Crazazy@feddit.nl 2 months ago
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