Mapping a Vector to a Vector Preserving Operations

[RyozKidz]

can i know how to map a vector to a vector by preserving the operation if addition and mutiplication ..pls dun use f(x+y)=f(x)+f(y)..
i wan to know how to use in abstract ...
if i do the mapping wat will happens?

 

[Werg22]

Do you mean an isomorphism where addition and the inner-product are preserved?

 

[HallsofIvy]

can i know how to map a vector to a vector by preserving the operation if addition and mutiplication ..pls dun use f(x+y)=f(x)+f(y)..
i wan to know how to use in abstract ...
if i do the mapping wat will happens?

I have no idea what your question is! There are many ways to "map a vector to a vector", some linear, others not. Are you specifically talking about linear mappings? What do you mean by "pls dun use f(x+y)= f(x)+ f(y)"? I can interpret that as "please don't use f(x+y)= f(x)+ f(y)" but what's the point in talking about linear mappings if you don't use their basic properties? And, finally, what in the world do you mean by "what will happen"?

 

[RyozKidz]

i wan to know wat is 2 vector is under the linear mapping?
will it become a new vector?
actually how to preserve the addition or mutiplication operation when it is under
linear mapping??

 

[Office_Shredder]

The definition of a linear mapping is a map that preserves addition and scalar multiplication, so it doesn't make much sense to ask how you preserve that under a linear map.

A linear map takes vectors to vectors, but not necessarily in the same vector space

I hope this answers your questions; it's not really clear what you're confused about