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Thread: MATH2601 Higher Linear Algebra

  1. #51
    Ancient Orator leehuan's Avatar
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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by InteGrand View Post
    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 o.O

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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post
    Where does the inspiration come from that it just happens to be the image that satisfy this criteria o.O
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    Re: MATH2601 Higher Linear Algebra






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    Re: MATH2601 Higher Linear Algebra

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    Re: MATH2601 Higher Linear Algebra

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    Re: MATH2601 Higher Linear Algebra

    Woah, how do you do it so fast?

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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by boredofstudiesuser1 View Post
    Woah, how do you do it so fast?
    He's a machine!

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    Ancient Orator leehuan's Avatar
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    Re: MATH2601 Higher Linear Algebra

    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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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post
    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
    _______________

    Yes
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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post
    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
    _______________

    It's ok, we'll call you a machine too if it makes you feel better.
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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post
    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
    I don't know if this was the reason why, but in the Q you typed above, there's a typo (top-right entry should have 6 rather than 5 in the square root).
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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by InteGrand View Post
    I don't know if this was the reason why, but in the Q you typed above, there's a typo (top-right entry should have 6 rather than 5 in the square root).
    Oops. Nah I think that was just a typo as I typed it on the forums

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    Ancient Orator leehuan's Avatar
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    Re: MATH2601 Higher Linear Algebra

    This one's a bit long...





    Proven in i): 0 is the only eigenvalue of B (so B is nilpotent)




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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post
    This one's a bit long...





    Proven in i): 0 is the only eigenvalue of B (so B is nilpotent)





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    Ancient Orator leehuan's Avatar
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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by InteGrand View Post


    Oh of course. Once I drew out the Jordan chain again and looked carefully at what the question gave iii made sense.
    _________________________________________






    Tools permitted if useful: Binomial theorem for matrices that commute in multiplication, Cayley-Hamilton theorem

    Edit: Thanks IG I just saw where your reply was :P
    Last edited by leehuan; 14 Jun 2017 at 9:11 PM.

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    Re: MATH2601 Higher Linear Algebra

    No more questions for this sem after tomorrow.
    __________________


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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post
    No more questions for this sem after tomorrow.
    __________________

    Hint: Use the axioms to show that a + a0 = a.
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    Re: MATH2601 Higher Linear Algebra

    This is a highly open-ended question and everyone's opinion might be different.

    What's the easiest proof (or would be a very easy proof) of the Cauchy-Schwarz inequality to memorise?

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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post
    This is a highly open-ended question and everyone's opinion might be different.

    What's the easiest proof (or would be a very easy proof) of the Cauchy-Schwarz inequality to memorise?
    Well you wrote one up here before, so maybe you'd find that easiest to "memorise" for yourself:

    Quote Originally Posted by leehuan View Post









    I did not even know that there was a sum form until doing past papers for 1251. Then I had to figure out why the sum and vector forms were equivalent.
    Note that it needs to be adapted slightly to deal with the complex case, but it's not too big a deal.

    You can also probably find many proofs online. There are twelve proofs here, but they seem to only be for the case of R^n: http://www.uni-miskolc.hu/~matsefi/O...rticle1_19.pdf .
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    Re: MATH2601 Higher Linear Algebra

    Completely forgot about that one.
    _________________

    Last edited by leehuan; 23 Jun 2017 at 8:29 PM.

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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post
    Completely forgot about that one.
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    Last edited by InteGrand; 24 Jun 2017 at 10:21 PM. Reason: TeXified
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    Ancient Orator leehuan's Avatar
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    Re: MATH2601 Higher Linear Algebra

    This is just some personal fun

    Last edited by leehuan; 19 Jul 2017 at 6:39 PM. Reason: Oh right, my bad

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    Re: MATH2601 Higher Linear Algebra

    Quote Originally Posted by leehuan View Post
    This is just some personal fun

    Yes. (I assume you meant the field to be R.)
    Last edited by InteGrand; 19 Jul 2017 at 1:57 PM.

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    Re: MATH2601 Higher Linear Algebra




































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    Re: MATH2601 Higher Linear Algebra

    Hopefully you don't mind if i post a question here. (taking MATH2601 this semester)

    Suppose that G is a group with precisely three distinct elements e (the identity), a and b.
    a) Prove that ab = e (Hint: eliminate other possibilities).
    b) Prove that a^2 = b.
    c) Deduce that G = {e, a, a^2} and hence that G is isomorphic to the group.

    (How do you get LaTeX to work here?)
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