Solving Matrix Basis Problem with Orthogonal Matrix B | Need Help Urgently

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The discussion revolves around a homework problem involving two bases and an orthogonal matrix B, where the goal is to prove that the lambdas are equal. One participant expresses difficulty in expressing vectors in terms of the given bases and seeks assistance. Another contributor clarifies that the lambdas can only be equal if the two bases are the same, implying that the proof may not require showing equality. They suggest focusing on properties preserved by orthogonal matrices, such as vector norms and inner products, which may be key to solving the problem. The conversation emphasizes the need for clarification on the problem statement and the correct approach to the proof.
Lindsayyyy
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Hi everyone

Homework Statement



File at attachment. Given are two basis and the orthogonal matrix B. When r=...(see attachment) I shall proof that the lambdas are equal.

Homework Equations



-

The Attempt at a Solution



I have much trouble with this exercise and it is quite urgent. I tried to express v1' via v1 and v2, but this doesn't bring me to the solution, for example I have: v1' = av1 + bv2 etc.

Can anyone help me with this?

Thanks for the help in advance
 

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Hi Lindsayyyy! :smile:

I don't really get your problem statement.

Any vector r can be represented uniquely with respect to a basis.
With respect to a different basis the representation is again unique, but will always be different.
So as I understand your problem, you can only proof that the lambdas are different.

The lambdas will only be the same iff the 2 basis are the same (that is, if M is the identity matrix).So I suspect you're not supposed to proof the lambdas are the same.
Especially seeing the last equation saying something about what appears to be the vector norm of an inner product of the lambdas.
Still not quite sure what it says though. Can you clarify?

I can say that vector norm and inner product are preserved by an orthogonal matrix, so you probably need to do something with that.
 
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