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MATH2601 Higher Linear Algebra (1 Viewer)

davidgoes4wce

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Re: MATH2601 Linear Algebra/Group Theory Questions

I don't know much about Graph Theory and Group Theory but are they two different topics? Or Different names but same subjects?

I can't be bothered Googling it
 

InteGrand

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Re: MATH2601 Linear Algebra/Group Theory Questions

I don't know much about Graph Theory and Group Theory but are they two different topics? Or Different names but Two dame subjects?

I can't be bothered Googling it
Two different topics.
 

leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions

I was wondering if given the trace and the determinant of a matrix could you write down a unique matrix satisfying these conditions, or a simple formula for the family of matrices satisfying it?

Mostly asking for the 2x2 case
 

InteGrand

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Re: MATH2601 Linear Algebra/Group Theory Questions

I was wondering if given the trace and the determinant of a matrix could you write down a unique matrix satisfying these conditions, or a simple formula for the family of matrices satisfying it?

Mostly asking for the 2x2 case
No, the value for the trace and determinant of a real or complex matrix does not uniquely specify the matrix.



(Note that for a 2x2 complex matrix, the trace and determinant will uniquely specify the eigenvalues of the matrix though.)
 
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leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions

The intuition of the question was this



I then realised that B may share the same eigenvectors as A, and have eigenvalues equal to the square root of those of A. But I'm not sure where to proceed from there.
 

InteGrand

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Re: MATH2601 Linear Algebra/Group Theory Questions

The intuition of the question was this



I then realised that B may share the same eigenvectors as A, and have eigenvalues equal to the square root of those of A. But I'm not sure where to proceed from there.
 

leehuan

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Re: MATH2601 Linear Algebra/Group Theory Questions



 

InteGrand

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Re: MATH2601 Linear Algebra/Group Theory Questions

No it need not be. Say V = R^2 and W1 = R^2 (= V) and W2 be the line (t, 0) (the x-axis). Take T to be a rotation map by 90 degrees counter-clockwise about the origin say. Then T is a linear map from V to V, so T(V) = T(W1) is a subspace of W1 = V = R^2, and W2 is a subspace of W1 (which is a subspace of V), but clearly W2 is not invariant under T (e.g. the point (1, 0) in W1 does not get mapped to a point in W2 by T; it gets mapped to (0, 1)).
 
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leehuan

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My approach thus far: Write



Is this a dead end? Because I don't see how I can use what I know about T here
 
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leehuan

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I feel bad lol. I had the same idea as InteGrand, I just mucked up my matlab input when I went to check my answer
_______________

 

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