MHB Need Assistance - Can Someone Help Me?

[FilipVz]

Hi,

can somebody help me with the following problem:View attachment 1530

Thank you. :)

 

[Sudharaka]

Hi,

can somebody help me with the following problem:View attachment 1530

Thank you. :)

Hi Filipvz, :)

Welcome to MHB! :)

You can find the proof for this theorem >>here<<.

 

[Deveno]

Lazy, hazy proof:

An isomorphism is, among other things, a bijection. So all one needs to do is show 2 things:

1) A linear injection preserves linear independence
2) A linear surjection preserves spanning

These two facts together show that the image under our given isomorphism of a basis for the first vector space is a basis for the second space, and since the isomorphism is bijective, they have the same cardinality.