• chonglibloodsport@lemmy.world
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      6 months ago

      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).

      • i_love_FFT@lemmy.ml
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        6 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 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.