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Rank and nullity

  1. Mar 20, 2014 #1
    1. The problem statement, all variables and given/known data

    Given that A is an 13 × 11 matrix of rank 3.

    a)The nullspace of A is a d-dimensional subspace of Rn.

    What is the dimension, d, and n of Rn?
    b) Is it true that for every b ∈ R11, the system ATx = b is consistent? (What does this means?)

    3. The attempt at a solution

    The row space has d = 3, n = 11

    The col space has d =3, n = 13

    The rows aren't linearly independent. The columns aren't linearly independent.

    The nullity of A is 8. (n =11; 11 - rank 3 = 8)

    a)

    The row space has d = 3, n = 11

    The col space has d =3, n = 13

    The rows aren't linearly independent. The columns aren't linearly independent.

    The nullity of A is 8. (n =11; 11 - rank 3 = 8)

    How do I find d and n of Rn?
     
    Last edited: Mar 20, 2014
  2. jcsd
  3. Mar 20, 2014 #2
    This question can be a little more easy if you examine some definitions.

    So, the basis must be a linearly independent set that spans a subspace.

    In the case of the rank of a matrix, this is defined as the dimension of your columnspace.

    If you columnspace has a dimension of 3, that means that you removed 8 other columns from your columnspace in order to make it an independent set. This means that you have 8 free variables in this matrix's reduced form.

    From this claim, you know that the nullspace is formulated based on the amount of free variables in your set. In fact, there is a theorem proving that the dimension of your nullspace is equal to the amount of free variables in a set of vectors.

    So what can you say about your nullspace's dimension based off of this?

    Also, there is a theorem stating that the subspace spanned by a set vectors of is determined by the amount of columns with a pivot. So, what can also be determined by this?
     
  4. Mar 20, 2014 #3

    Just to be sure, nullspace = nullity?

    When you talk about set, are you referring to the columnspace or both the row space and column space?
    "From this claim, you know that the nullspace is formulated based on the amount of free variables in your set. In fact, there is a theorem proving that the dimension of your nullspace is equal to the amount of free variables in a set of vectors. "


    null(A) = 11 - 3 = 8; hence, 8 free variables in the set of solution for which Ax = 0
    null(AT) = 13 - 3 = 10; 10 free variables in the set of solution for which A Tx = 0


    There are 8 free variables in the row space and 10 free variables in the column space.

    Edit: I think there are 2 types of ''nullspace''

    dimension of column space and dimension of nullspace, the sum of which = dimension of Rn. The question ask for the nullspace of A and nullspace is simply the number of free variables in the row space of A. 8 free variables, hence nullspace(A) = 8

    They matrix A is an 13 x 11 matrix. n = 11.

    Therefore, the dimension, d, and n of Rn is such that d = 8 and n = 11
     
    Last edited: Mar 20, 2014
  5. Mar 20, 2014 #4
    I'm unclear as to which theorem states that subspace spanned by a set vectors of is determined by the amount of columns with a pivot. Is there a name for this theorem?
     
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