Science Memes

10940 readers

1988 users here now

Welcome to c/science_memes @ Mander.xyz!

A place for majestic STEMLORD peacocking, as well as memes about the realities of working in a lab.

Rules

Don't throw mud. Behave like an intellectual and remember the human.
Keep it rooted (on topic).
No spam.
Infographics welcome, get schooled.

This is a science community. We use the Dawkins definition of meme.

Research Committee

[email protected]

Other Mander Communities

Science and Research

Biology and Life Sciences

Physical Sciences

Humanities and Social Sciences

Practical and Applied Sciences

Memes

Miscellaneous

founded 2 years ago

MODERATORS

[email protected]

504

Explain yourselves, comp sci. (mander.xyz)

submitted 6 months ago by [email protected] to c/[email protected]

71 comments fedilink hide all child comments

you are viewing a single comment's thread
view the rest of the comments

[–] [email protected] 10 points 6 months ago* (last edited 6 months ago) (1 children)

Your polynomial, f(x) = a + bx +cx^2 + dx^3, is an element of the vector space P3(R), 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(R) 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).

[–] [email protected] 1 points 6 months ago* (last edited 6 months ago) (2 children)

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^2^x^6^, 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.

[–] [email protected] 3 points 6 months ago* (last edited 6 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 nor the bilinear product of an algebra.

[–] [email protected] 1 points 6 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