linear transformation from R n </msup> to R m </msup> can be

Brooke Webb

Brooke Webb

Answered question

2022-05-24

linear transformation from R n to R m can be represented in a matrix form. What about a transformation from a:
1) Infinite dimension to infinte dimension
2) finite to infinite dimension
3) infinite to finite dimension Can they represented by matrix form....? Before this another question is there a linear transformation from Infinite dimension to infinte dimension, finite to infinite dimension and vice versa..?

Answer & Explanation

Maximo Sweeney

Maximo Sweeney

Beginner2022-05-25Added 7 answers

As to matrix representation, finite-finite version is evident. In general case, as long as range or domain is infinite dimensioned, it's impossible to build a matrix. The key problem is that matrix form depends on basis in both domain and range of the application. In general linear space of infinite dimension we know the existence of a Hamel basis if we accept the axiom of choice.
Sometimes in some simple cases we can say that a certain linear operator T can be viewed as a matrix (i.e. we know basis in both range ( r j ) and domain ( d i ), and we can develop T d i in the basis r j ), but again, in general case mathematicians prefer just to think about linear operator and not its matrix form.

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