Need Assistance - Can Someone Help Me?

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In summary, the conversation is about a person asking for help with a problem and being directed to a proof for a theorem on an online forum. The proof involves showing that a linear injection preserves linear independence and a linear surjection preserves spanning. This proof demonstrates that the image of a basis under the given isomorphism is also a basis, thus proving the theorem.
  • #1
FilipVz
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Hi,

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

Thank you. :)
 

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  • #2
FilipVz said:
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<<.
 
  • #3
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.
 

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