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Some linear algebra proofs I couldn't figure out: Help!

  1. Jun 1, 2005 #1
    I am "scared" (to put it mildly) of these problems, which I need to review before my final tomorrow. Just to let all of you know, this is not homework. There are 25 or so problems, and I only understand around 10 of them. :frown:
    Help me! I need an A on the final to get a B in the class. :cry:

    1. If I: W-->W is the identity linear operator on W defined by I(w) = w for w in W, prove that the matrix of I repect with to any ordered basis T for W is a nXn I matrix, where dim W= n

    2. Let L: W-->W be a linear operator defined by L(w) = bw, where b is a constant. Prove that the representation of L with respect to any ordered basis for W is a scalar matrix.

    3. Let X,Y, Z be sqaure matrices. Show that: (a) X is similar to Y. (b) If X is similar to Y then Y is similar to X. (c) If X is similar to Y and Y is similar to Z, then X is similar to Z.

    4. Show that if X and Y are similar matrices then X^k and Y^k are similar for any positive integer k.

    5. Show that if X and Y are similar, then transpose(X) and transpose(Y) are similar.

    6. X and Y are similar. Prove: 1) If X is nonsingular, then Y is nonsingular. 2) If X is nonsingular, then A^-1 and B^-1 are similar.

    7. Prove that A and transpose(A) have equivalent eigenvectors. How are the associated eigenvectors of A and transpose(A) related?

    8. Prove that if A^k = O for some positive k (A is a nilpotent matrix then, right?) then 0 is the only eigenvalue of A.

    In 9, X is a mxm matrix
    9a. Show that det(A) is the product of the roots of the characteristic polynomial of A.
    9b. Show that A is singular if and only if 0 is an eigenvalue of A.
    9c. Prove that if L: V--->V is a linear transformation, show that L is not one-to-one if and only if 0 is an eigenvector of L.

    10. Show that if A and B are orthogonal matrices, then AB is a orthogonal matrix.

    11. Show that if A is an orthogonal matrix, then det(A) is +/- 1.

    Please everyone, I know that you may be annoyed at me for posting so many problems, but my final is tomorrow, and I am getting mental blocks on these problems. So, please, PF Members, will you also show your work in detail, so I can follow it? There's no solution pack for my review WS.
    Again, detailed proofs would be really appreciated. I am so stressed about the final. :surprised
     
    Last edited: Jun 1, 2005
  2. jcsd
  3. Jun 1, 2005 #2
    just scanning, 10 seems pretty easy: if [itex]A^T A = I[/itex], and [itex]B^T B = I[/itex], what is [itex] (AB) (AB)^T[/itex]?
    as for 11, use the property of determinants that [itex]|AB|=|A||B|[/itex] on the above relation.
     
    Last edited: Jun 1, 2005
  4. Jun 1, 2005 #3
    Isn't it (A^-1) transpose(A) = I, not (A)transpose(A) = I?
    But thanks.
    More help, anyone?
     
  5. Jun 1, 2005 #4

    mathwonk

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    the first i recommend is learn to deal with hatred. maybe anger management classes would help.
     
  6. Jun 1, 2005 #5
    no, an orthogonal matrix's transpose is the inverse of the matrix, so [itex]A^{-1}=A^T[/itex].
     
  7. Jun 1, 2005 #6
    DarkEternal....Oh, I see..thanks

    mathwonk....not really mad, just fine actually. More frustrated than anything. Care to help?
     
  8. Jun 1, 2005 #7
    Please, help me. I can't solve any of them.
    Final tomorrow. Have mercy! :cry: , PF members!
     
  9. Jun 1, 2005 #8

    mathwonk

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    get a grip. read the definitions of the absic concepts, such as the matrix of a linear transformation wrt a given basis. then you should be able to do #1. don't give up until you see how to do it. everyone has the same angst, "help, I can never do this....ahhhh..."

    but not everyone gives in to it as totally as you are doing.

    hang in there ,, you are a smart person, and what one person can do another can.
     
  10. Jun 2, 2005 #9

    matt grime

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    What is the definition of "similar"? If you write out what it means for X to be similar to Y, then just simple algebra will prove 3,4,5,6

    What do you know about characteristic polynomials? In any case, for 8, if t is an evalue of A then Av=tv for some v, right? What does that say about repeatedly applying A?
     
  11. Jun 2, 2005 #10
    the matrix for an operator with respect to some basis has columns given by the value of the operator on each basis vector, written with respect to that basis. for each basis vector e_i, I(e_i) is e_i.

    Same as above, only now I(e_i) = b*e_i

    I assume you mean X is similar to X, for part (a). Otherwise the statement is not true.

    (a) X=IXI^-1
    (b) if X=AYA^-1, then Y=A^-1XA
    (c) if X=AYA^-1 and Y=BZB^-1, then X=(AB)Z(AB)^-1


    (AXA^-1)^k = AX^kA^-1

    (AXA^-1)^T=(A^T)^-1X^TA^T

    det(AXA^-1)=det(X)

    if Au=xu, then [itex](A\mathbf{u})^\dagger=\mathbf{u}^\dagger A^\dagger=(x\mathbf{u})^\dagger=\overline{x}\mathbf{u}^\dagger[/itex]

    if Au=xu, then A^ku=x^ku

    The sum of the roots of a polynomial over an algebraically closed field is equal to the constant term of the polynomial
    this follows from 9a and the definition of singular
    If V is finite dimensional, then any one-to-one linear map is nonsingular. More generally, if L is not injective, then there are some distinct u and v with L(u)=L(v), and then u-v has eigenvalue 0
    (AB)^T(AB)=A^TB^TBA

    det(A^T)=det(A)
     
    Last edited: Jun 2, 2005
  12. Jun 2, 2005 #11

    mathwonk

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    has it occurred to you that you do not deserve an A, or even a B, based on your level of understan`ding?
     
  13. Jun 3, 2005 #12
    I'm inclined to agree.
     
  14. Jun 3, 2005 #13
    "Let's all take turns patting each other on the back for being smarter than everyone else. Yep, it feels good."
     
  15. Jun 3, 2005 #14

    mathwonk

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    actually one usually has to take some flack for telling people the truth on these threads. the point is that it is sadly out of style to tell people that understanding the subject is more important than handing in other peoples work to get a grade they do not deserve. if one of them wakes up and begins to actually do some of their own work, we hope that will hopefully help them in the long run. eventually maybe people here will begin to appreciate being advised to stand on their own feet.
     
    Last edited: Jun 3, 2005
  16. Jun 3, 2005 #15
    I totally agree. But couldn't you give them that advice without sounding so condescending? You do this all the time and it makes me wonder if you are using this forum to relieve stress at others' expense.
     
  17. Jun 3, 2005 #16

    mathwonk

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    so are you are saying that even true advice is useless if given undiplomatically? you may be right. But in my own life some blunt advice has helped after I thought about it.
     
    Last edited: Jun 3, 2005
  18. Jun 3, 2005 #17
    Yes, most everything you say is true (even the undiplomatic stuff). But when it is that harsh, it defeats your noble purpose, which is to help them to become better at math. Having someone tell you "it's obvious to me, what don't you understand?" doensn't help much when you honestly have no clue. And it doesn't make you want to learn, either.

    BTW, I understand you don't get paid to help people here. You really can say whatever you feel like saying.

    edit: no fair, you keep editing your post so it makes what I said look invalid. Make up your mind.
     
    Last edited: Jun 3, 2005
  19. Jun 3, 2005 #18

    mathwonk

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    you are being honest. I have no problem with that. I admit to being tired and stressed many times. If I were infinitely energetic and patient I would be a better teacher. perhaps i will logoff until i get some more rest.

    i admit i have trouble with:

    1) self appointed experts who do not understand the topic at all, but who are nonetheless dogmatic in their (misguided) statements. These people are misleading the students, hence doing harm.

    2) students who beg someone else to do their work for them, and blame their professor for all their problems. these people are their own worst enemies and are harming themselves. I do not know how to help them, and usually I avoid their posts but sometimes I try to give them a wakeup call.


    The idea that someone logs on here and asks someone else to do their work so they can hand it in to some trusting professor who will then waste hours grading my work or your work instead of his student's work, is anathema to me.

    I think I am seldom impatient with anyone actually interested in learning, who asks for genuine help, not gifts or solutions, and who is willing to do some actual labor.

    Of course I may be wrong. I have many faults.
     
    Last edited: Jun 3, 2005
  20. Jun 3, 2005 #19

    mathwonk

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    so I guess i am saying: my credo is:

    1) tell the truth and

    2)be willing to do your own work.

    anyone who accepts these groundrules will not be criticized by me.
     
  21. Jun 3, 2005 #20

    mathwonk

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    forgive me for editing my posts. it was not to invalidate your responses, which are very intelligent and fair. When something just flies off my keyboard, I often reconsider and go back and edit it, but sometimes someone is already offended. My apologies. I am actually trying to teach people from the background of my own experience, some of which is that they may need to change their approach to learning.

    I myself never got anywhere until i stopped blaming all my teachers for my problems. I also think it helped me that my college teachers actually gave me D's and F's when I earned them. One professor in grad school I still respect greatly, gave me "no credit" in a course where i did so little that he could not in good conscience give me any grade at all.

    That apparently seldom happens today. Many students who do and learn essentially nothing still receive grades, even A's, in courses from which they carry away no knowledge at all. This is a disservice to those students who will face a sad reckoning when they try to earn a living.

    that is why when someone logs on here and says "I need to get an A, please work my final for me", rather than "I need to understand linear independence, please explain it to me," I may react negatively.

    thank you however for the reminder to at least try to practice as much diplomacy as possible. surely you are right.

    by the way, when you criticize me both for posting harsh comments and also for editing them out, do you think you might possibly be just a tad hard to please?
     
    Last edited: Jun 3, 2005
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