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s3a

  • 2 years ago

Question: If A is a 3 by 5 matrix, what information do you have about the nullspace of A? Answer: N(A) has dimension at least 2 and at most 5. My trouble: How do we know this?

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  1. ingenuus
    • 2 years ago
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    m>=r m=3 so 0=<r=<3 dim N(A) = n-r=5-r 2<=5-r<=5

  2. s3a
    • 2 years ago
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    You're using the notation where m is the number of rows and r is the rank, right?

  3. ingenuus
    • 2 years ago
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    yes m=#rows n=#columns r=rank

  4. s3a
    • 2 years ago
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    So m >= r always holds for the nullspace of a matrix?

  5. ingenuus
    • 2 years ago
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    rank is #pivot rows right? can it be more than #of rows?

  6. ingenuus
    • 2 years ago
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    it is just a fact for every matrices

  7. ingenuus
    • 2 years ago
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    i mean you can't have more pivot 'rows' than rows

  8. s3a
    • 2 years ago
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    No, it cannot. I now get the answer to that sub-question.

  9. s3a
    • 2 years ago
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    I mean my sub-question not part (b). Let me look at 3(b) again

  10. s3a
    • 2 years ago
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    Why is dim N(A) = n-r=5-r?

  11. s3a
    • 2 years ago
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    Also is dim N(A) = dim ("the x variable vector in Ax = 0")?

  12. ingenuus
    • 2 years ago
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    i don't know whether you've heard the mit ocw lecture or not, but you should know dim N(A) = n -r

  13. ingenuus
    • 2 years ago
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    matrix is 3by 5 so n=5

  14. s3a
    • 2 years ago
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    I haven't. Is N(A) = n - r something I am supposed to "just memorize"?

  15. s3a
    • 2 years ago
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    dim N(A) = n - r that is

  16. ingenuus
    • 2 years ago
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    well dim N(A) is number of vectors in basis of nullspace of A

  17. ingenuus
    • 2 years ago
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    think about the basis of nullspace of A think how you get those bases

  18. s3a
    • 2 years ago
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    (where n and r are of the matrix A) (Sorry for the potentially dumb question, I'm kind of braindead at the moment but, like I said, I cannot take a break.)

  19. s3a
    • 2 years ago
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    (My break will be when I sleep.)

  20. ingenuus
    • 2 years ago
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    can you think how you obtain the null space of A?

  21. ingenuus
    • 2 years ago
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    then it is clear cut that dim N(A) = n-r

  22. s3a
    • 2 years ago
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    you find the vector x in Ax = 0

  23. s3a
    • 2 years ago
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    ?

  24. ingenuus
    • 2 years ago
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    yeah i'm asking do you know the exact procedures

  25. s3a
    • 2 years ago
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    get A^(-1) and multiply both sides?

  26. s3a
    • 2 years ago
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    (on the left)

  27. s3a
    • 2 years ago
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    i mean left multiplication on each side of the equation

  28. ingenuus
    • 2 years ago
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    since there are r pivot columns, there are n-r free columns and there are n-r special solutions to the null space. And they form basis for nullspace. so dim N(A)=n-r

  29. ingenuus
    • 2 years ago
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    you can do that only when a is invertible

  30. ingenuus
    • 2 years ago
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    you use elimination process to compute nullspace right?

  31. ingenuus
    • 2 years ago
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    why don't you solve some examples to get dimN(A)=n-r i think you know the definitions

  32. s3a
    • 2 years ago
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    when a is invertible, like you said. I'm not grasping something fundamental though: what is the nullspace of A? is it x in Ax = 0? In other words, what object's dimension is n - r?

  33. ingenuus
    • 2 years ago
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    nullspace(A) is a subspace in R^n, which a vector in it satisfies Ax=0

  34. ingenuus
    • 2 years ago
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    it is just whole solutions to Ax=0

  35. ingenuus
    • 2 years ago
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    solutions form a subspace, so they are called null'space'

  36. s3a
    • 2 years ago
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    ok so it's a dimension of a vector space in which a vector x makes Ax = 0?

  37. ingenuus
    • 2 years ago
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    for example, when A is invertible, x is only 0 and the N(A)={0}

  38. ingenuus
    • 2 years ago
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    no nullspace is just a name of subspace

  39. ingenuus
    • 2 years ago
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    dim N(A) is what you are saying

  40. s3a
    • 2 years ago
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    ya for two seconds, my brain was thinking about dim N(A).

  41. s3a
    • 2 years ago
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    Will you be here in 1..75 hours?

  42. ingenuus
    • 2 years ago
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    you need to carry on the elimination yourself to see dim N(A) = n-r

  43. s3a
    • 2 years ago
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    I have to go eat now.

  44. ingenuus
    • 2 years ago
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    i'm sorry

  45. s3a
    • 2 years ago
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    (If you keep writing, I will come back and read what you said.)

  46. ingenuus
    • 2 years ago
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    i have to go :(

  47. s3a
    • 2 years ago
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    o :(

  48. ingenuus
    • 2 years ago
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    i have a pdf about nullspace. do you want it?

  49. ingenuus
    • 2 years ago
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    http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces/solving-ax-0-pivot-variables-special-solutions/ download the lecture summary for that lecture. it has the just the right information for you. good luck

  50. s3a
    • 2 years ago
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    thanks and sry for leaving abruptbly

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