Vectors have a set of rules they need to obey (e.g. scalar addition/multiplication/etc)

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

The set of linear operators from a vector space V to a vector space W is itself a vector space.

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.

what about tensors?

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>what about tensors
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Also what about multivectors? Is that the point of multivectors - forming a group or other structure? What do multivectors bring that tensors do not have?

a dot product is a homomorphism from vector space to metric space, so no.

>mfw today I discovered metric space is a thing

>homomorphis between vector space and metric space
Black nigger what are you on.

metric space is a set with a distance defined between points of the set

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

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.

Fields have addition and mult.
Vector spaces over fields add the operations of vector addition and scalar mult.

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...

>abstract algebra

'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.

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.

elaborate on what you mean by
>A field can be viewed as a kind of space

Not catching your drift on that part

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.

Whoops, I should have said field axioms.

thanks annon

who is she?

i'm not talking about the cross product, which is why i said "dot product"

someone who is out of your league

Well aware of that. My post remains fully correct.

>mfw this is the only thread on Veeky Forums involving intelligent discussion of any sorts

...

it is a fact that pedophiles are marginally make up the smarter percent of the population