Not always. Any m by n matrix is also a vector. Polynomials are vectors. As are continuous functions.
A vector is an element of a vector space over a field. These are sets which which a few operations, vector addition and scalar multiplication, and obey some well known rules, such as the existence of a zero vector (identity for vector addition), associativity and commutativity of vector addition, distributivity of scalar multiplication over vector sums, that sort of thing!
These basic properties give rise to more elaborate concepts such as linear independence, spanning sets, and the idea of a basis, though not all vector spaces have a finite basis.
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 queen written as a linear combination of the basis vectors with real valued scalars.
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.
Every vector is a tensor. Matrices are vectors. Magnitude and direction have nothing to do with the definition of vectors which are just elements of vector spaces.
All vectors are tensors but not vice versa. And every page/definition of vector I’ve seen references magnitude and direction, even the vector space page you linked.
It looks like “vector” commonly refers to geometric vectors which is what most folks in this thread are discussing.
Would N by M vectors be imaginary, where each DOF has real and imaginary components?
chonglibloodsport@lemmy.world 5 months ago
Not always. Any m by n matrix is also a vector. Polynomials are vectors. As are continuous functions.
A vector is an element of a vector space over a field. These are sets which which a few operations, vector addition and scalar multiplication, and obey some well known rules, such as the existence of a zero vector (identity for vector addition), associativity and commutativity of vector addition, distributivity of scalar multiplication over vector sums, that sort of thing!
These basic properties give rise to more elaborate concepts such as linear independence, spanning sets, and the idea of a basis, though not all vector spaces have a finite basis.
muntedcrocodile@lemm.ee 5 months ago
How are polynomials vectors how does that work?
Say u have polynomial f(x)= a + bx + cx^2 + dx^3
How is that represented as a vector? Or is it just one of those maths well technically things? Cos as far as I’m aware √g = π = e = 3.
Are differential eqs also vectors?
chonglibloodsport@lemmy.world 5 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 queen written as a linear combination of the basis vectors with real valued scalars.
i_love_FFT@lemmy.ml [bot] 5 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.
Pulptastic@midwest.social 5 months ago
Wouldn’t N by M be a tensor? Magnitude and direction only need one entry per DOF.
chonglibloodsport@lemmy.world 5 months ago
Every vector is a tensor. Matrices are vectors. Magnitude and direction have nothing to do with the definition of vectors which are just elements of vector spaces.
Pulptastic@midwest.social 5 months ago
All vectors are tensors but not vice versa. And every page/definition of vector I’ve seen references magnitude and direction, even the vector space page you linked.
It looks like “vector” commonly refers to geometric vectors which is what most folks in this thread are discussing.
Would N by M vectors be imaginary, where each DOF has real and imaginary components?