vectors have a set of rules they need to obey (e.g. scalar addition/multiplication/etc)
Do matrices/tensors also have a set of governing rules?
>plz help
>>8097375
The set of linear operators from a vector space V to a vector space W is itself a vector space.
>>8097375
yes and no. "matrices" are not a group. it's easier if you restrict yourself to a set of matrices that all obey the same rules. for example, you can solve many problems if you only consider the set of NxN invertible matrices. those obey a similar set of rules as the ones you described.
>>8097401
what about tensors?
>>8097403
a dot product is a homomorphism from vector space to metric space, so no.
>>8097418
>mfw today I discovered metric space is a thing
>>8097418
>homomorphis between vector space and metric space
Black nigger what are you on.
>>8097428
metric space is a set with a distance defined between points of the set
>>8097448
in laymans terms, whats the difference between a space and a field?
(e.g. scalar field/vector field) vs (vector space/metric space)
Im not looking for rigor, just some intuition
>>8097375
You sir, should study abstract algebra.
There are many ways to represent mathematical objects using matrices and tensors, and that subject covers a plethora in great detail.
>>8097453
Fields have addition and mult.
Vector spaces over fields add the operations of vector addition and scalar mult.
>>8097439
idiocy. it's not a homo, but it does take elements from a vector space to a metric space - unless i'm still an idiot...
>>8097459
>abstract algebra
>>8097453
'Space' is more or less just a synonym for a set, except that you typically only call a set a space if it also has some extra 'structure' (think rules or axioms). A metric space is a set where the objects have a metric function defined on them. A topological space is a set where the set has a topology defined on then. A vector space is a space that obeys the axioms for a vector space. A field can be viewed as a kind of space. It is also possible for a set to be many different types of spaces at the same time. For instance the reals are a metric space, a topological space, a vector space, and a field.
>>8097463
Well, you're definitely an idiot in any case. Aside from that, the metric on R doesn't really play a role here. What matters is the ordering and the structure of R as a vector space over itself. Now what you have is a bilinear map from V x V to R, which is the same as a linear map from the tensor product V(x)V (which is again a vector space) to R. So if anything, you have a homomorphism between vector spaces.
>>8097479
elaborate on what you mean by
>A field can be viewed as a kind of space
Not catching your drift on that part
>>8097486
A field is a set that also obeys the ring axioms. This restriction is said to give it structure. So really a field is a space as well.
That said, I believe it may be unusual to actually call a field a space, not because it's wrong but because it's just uncommon.
>>8097512
Whoops, I should have said field axioms.
>>8097514
thanks annon
>>8097375
who is she?
>>8097481
i'm not talking about the cross product, which is why i said "dot product"
>>8097533
someone who is out of your league
>>8097540
Well aware of that. My post remains fully correct.
>mfw this is the only thread on /sci/ involving intelligent discussion of any sorts
>>8097545
>>8097581
it is a fact that pedophiles are marginally make up the smarter percent of the population
