MATH2601 Higher Linear Algebra (1 Viewer)

davidgoes4wce

Well-Known Member
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

Well-Known Member
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

Well-Known Member
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

Well-Known Member
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

leehuan

Well-Known Member
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

Well-Known Member
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

leehuan

Well-Known Member
Re: MATH2601 Linear Algebra/Group Theory Questions

InteGrand

Well-Known Member
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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• kawaiipotato and leehuan

leehuan

Well-Known Member

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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InteGrand

Well-Known Member

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
Claim: W := im(T) is such a subspace.

Proof: Exercise.

• kawaiipotato and leehuan

leehuan

Well-Known Member
Claim: W := im(T) is such a subspace.

Proof: Exercise.
Where does the inspiration come from that it just happens to be the image that satisfy this criteria InteGrand

Well-Known Member
Where does the inspiration come from that it just happens to be the image that satisfy this criteria • leehuan

He-Mann

Vexed?

• leehuan

boredofstudiesuser1

Active Member
Woah, how do you do it so fast?

leehuan

Well-Known Member
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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InteGrand

Well-Known Member
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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Yes

• leehuan

boredofstudiesuser1

Active Member
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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It's ok, we'll call you a machine too if it makes you feel better. • leehuan