MHB Need Assistance - Can Someone Help Me?

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The discussion revolves around a request for help with a mathematical problem related to isomorphisms in vector spaces. A user provides a link to a proof and outlines key concepts: a linear injection preserves linear independence, while a linear surjection preserves spanning. Together, these principles demonstrate that the image of a basis under an isomorphism maintains the same properties in both vector spaces. The bijective nature of the isomorphism ensures that both spaces have the same cardinality. The conversation emphasizes the importance of understanding these foundational concepts in linear algebra.
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Hi,

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

Thank you. :)
 

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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<<.
 
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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