Can someone with access to Shankar's QM book help me (vector spaces)?

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The discussion centers on a specific point in Shankar's Quantum Mechanics book regarding the expression 1|V⟩ = |V⟩ for all vectors |V⟩. It questions whether this relationship is an assumption or can be derived from the axioms presented in the text. Participants note that this property is often treated as an axiom in linear algebra, suggesting that it may not require proof. The conversation emphasizes the importance of understanding the foundational principles of vector spaces in quantum mechanics. Clarification on this point could enhance comprehension of the material.
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On pg. 2, Shankar seems to just assume (I'm guessing) that 1|V\rangle = |V\rangle for all vectors |V\rangle when he does the exercise at the bottom of the page. Is this true, or is it possible to prove 1|V\rangle = |V\rangle from the axioms he lists?
 
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I think that this is what the author means by "do what is natural". Usually, however, 1|v> = |v> is just listed as an axiom ("there is a scalar multiplication r|v> with 1|v> = |v> and 0|v> = |0>").
 

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