>a vector is something that has a magnitude and a direction

>>7962318
It's a true statement in a context.

>>7962318
But that's what it is.

>>7962323
No. Vector spaces are on fields, which extends to more than just the reals/complex numbers.

>engiNERD detected

>>7962326

To be fair, that's not what vectors meant at first; mathematicians generalized them later and didn't bother changing the name, but traditional 'vectors' should be considered a subset of 'generalized vectors'.

Well you Can define a norm for every vector space so that's that.

My college algebra professor defined a vector as a matrix with only one column.

A vector is something that transforms as a vector.

>>7962326
found the undergraduate

>>7962318
Who is this semen demon

>>7962318
>you can take the dot product between 2 vectors
when will this meme die? why cant we just learn about covariance and contravariance of vectors from the start?

A vector is a mathematical entity with n components in n space

>>7962570
thats a tuple you retard. the transformation rules make a vector

>>7962577
pls no bully

>>7962577
Lol you need to be told how many numbers are written there instead of simply counting them or using covariance laws!

>>7962318
>a vector is an element of a vector space
Literally autism

>>7962318
A vector is a spaceship in hyperspace

>>7962364
No you can't. Normed vector spaces are somewhat special. What is the norm on the space of all real functions?

>>7962318

When will this meme end?

When your mom stops being a scalar.

>>7962669
Pick a basis for it using your maths voodoo and set the norm as the maximum of the coefficients. This is well defined because every vector is a sum of finitely many basis vectors.

>>7962322
>It's a true statement in a context.

/thread

>>7962867
>This is well defined because every vector is a sum of finitely many basis vectors
lol no

A n-tuple is an object with n-components.

A vector is an element of a vector space.

A tangent vector at a point on a manifold is a derivation of the germ algebra of the manifold at that point.

A vector field on manifold is a smooth section of the tangent bundle of the manifold.

>>7963212
Corrections:
>A n-tuple is an object with n-components.
needs to be ordered
>A vector field on manifold is a smooth section of the tangent bundle of the manifold.
The section does not have to be smooth

>>7962440
I'm in HS m8 don't fucking call me that

>>7962440
?
Vector spaces are only defined on fields.
Were you thinking of modules?

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>>7962867
>This is well defined because every vector is a sum of finitely many basis vectors.
>there are people this stupid posting on /sci/ right now

An element of [math] \mathbb{N}^3 [/math] has magnitude and direction to but it's not a vector.

>>7963527
"A vector is something that has magnitude and direction" =/= "Everything with magnitude and direction is a vector"

>>7962891
>>7963523
That's the literal definition of a basis.

http://www.math.lsa.umich.edu/~kesmith/infinite.pdf

My tutor just says "a vector is an arrow"

good enough for me lads

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>>7963541
>finitely basis vectors
>links a PDF that demonstrates vector spaces with infinite dimension
What the fuck are you doing, retard?

A vector is an ordered set of elements of the field, over which the vector space is defined.

I like thinking about them as linear maps moving each point by adding to it's components.

>>7963555
>>7963523
>>7962891
Zorn's lemma implies the existence of a Hamel basis for every vector space, the basis may be infinite, but all vectors can be represented as a finite sum its elements, that's what he meant, you dumb shits.
https://en.wikipedia.org/wiki/Basis_(linear_algebra)#Proof_that_every_vector_space_has_a_basis

>>7963566
>all vectors can be represented as a finite sum its element
That's false lmao

>>7963571
The basis may be infinite, but that's the exact definition of a Hamel basis, you're probably thinking about Schauder basis.

>>7963571
There's not even a notion of infinite sums unless you already have a norm picked out.

>>7963630

I'm telling you that there's no infinite sum involved, that's the main difference between a Hamel basis and a Schauder basis, just look it up.

>>7963552
Keep the receipt.

>>7963678
I'm agreeing with you -- check whom I replied to.

>>7963678
are you confusing countable with finite?

>>7962867
not necessarily finite. All periodic cont signals with period T can be represented as linear combinations of e^(2*pi*n/T), n in Z(fourier series). that set of e^(2*pi*n/T) forms a basis for all periodic functions with period T.
I wonder if fourier transform also can considered in the vector space perspective, im not sure as the basis has uncountably many elements

https://en.wikipedia.org/wiki/Schauder_basis