/sci/ - Science & Math

[Anonymous No.7076927]

A VECTOR SPACE IS...

[Anonymous No.7076931]

A collection of vectors that have multiplication and addition properties defined such that it constitutes a vector space?

[Anonymous No.7076964]

>>7076931
Don't forget that it also must be closed under scalar multiplication and vector addition.

[Anonymous No.7076982]

>>7076927
a module over a field.

[Anonymous No.7076983]

A set closed under linear combinations.

[Anonymous No.7077024]

>>7076927
A triplet (V,+,*) s.t. (V,+) is an abelian group and.... etc.

[Anonymous No.7077030]

>>7076982
winrar

[Anonymous No.7077035]

>>7076927
An object in Vect_k, for some field k!!
(This was meant to be humorous, but seriously guys, there is an nLab page where they define a super vector spaces as objects in the category of super vector spaces over a field k... I mean, FUCK.)

[Anonymous No.7077060]

<span class="math">(\mathbb{V};+;\cdot)[/spoiler] with <span class="math">(\mathbb{V};+)[/spoiler] an abelian group, <span class="math">\cdot: \mathbb{K}\times\mathbb{V} \rightarrow \mathbb{V}[/spoiler] distributive over <span class="math">+[/spoiler], associative and <span class="math">\forall \mathbf{x} \in \mathbb{V}, 1\cdot \mathbf{x} = \mathbf{x}[/spoiler].

[Anonymous No.7077082]

>>7076982
>mfw seminar starts with equating certain modules with matrix algebras over division rings and then broadly discusses theorems about their endomorphism rings over themselves and I don't understand half of it

[Anonymous No.7077083]

>>7076982
>mfw seminar starts with equating certain modules with matrix algebras over division rings and then discusses theorems about their endomorphism rings and I don't understand half of it

[Anonymous No.7077096]

>>7077083
What parts aren't you following?

[Anonymous No.7077104]

>>7077096
I caught up. The lecturer referred me to a book on representation theory and now I can follow along okay.

[Anonymous No.7077114]

>>7077104
Oh, that's good. Representation theory is awesome, enjoy the ride.

[Anonymous No.7077135]

A point in <span class="math">\mathbb{\bold{R}}^{n}[/spoiler] is frequently also called a vector in <span class="math">\mathbb{\bold{R}}^{n}[/spoiler], because <span class="math">\mathbb{\bold{R}}^{n}[/spoiler], which <span class="math">x+y=(x^1+y^1,...,x^n+y^n)[/spoiler] and <span class="math">ax=(ax^1,...,ax^w)[/spoiler], as operations, is a vector space (over the real numbers, of dimension n).

[Anonymous No.7077142]

>>7077114
+1, one of major pillars of modern mathematics imo.

[Anonymous No.7077148]

A point in <span class="math">\mathbb{\boldsymbol{R}^{n}}[/spoiler] is frequently also called a vector in <span class="math">\mathbb{\boldsymbol{R}^{n}}[/spoiler], because <span class="math">\mathbb{\boldsymbol{R}^{n}}, which <span class="math">x+y=(x^1+y^1,...x^n+y^n)[/spoiler] and <span class="math">ax=(ax^1,...,ax^n)[/spoiler], as operations, is a vector space (over the real numbers, of dimension n).[/spoiler]

[Anonymous No.7077163]

>>7077142
Amen, I agree.

[Anonymous No.7077230]

the span of all linear combinations of a linearly independent set of vectors.

[Anonymous No.7077478]

>>7076927
the place where vectors live?

[Anonymous No.7077753]

>>7076927
An abelian group V and a field F with a function VxF->V that satisfies some distributive laws.

Alternatively, a module where the associated ring is a field.