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[Linear Algebra] Nullspace equals Column space

  1. Aug 31, 2009 #1
    1. The problem statement, all variables and given/known data

    Why does no 3 by 3 matrix have a nullspace that equals its column space?

    2. Relevant equations

    NA

    3. The attempt at a solution

    [tex]
    A =
    \begin{bmatrix}
    0 & 0 & 1 \\
    0 & 0 & 0 \\
    0 & 0 & 0
    \end{bmatrix}
    \]
    [/tex]

    [tex]
    C(A) =
    \begin{bmatrix}
    1 \\
    0 \\
    0
    \end{bmatrix}
    \]
    [/tex]

    Does not then N(A) = C(A)?
    I think I am missing something here.

    Thank you for your time.
     
  2. jcsd
  3. Aug 31, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I don't know what you are missing because I don't know what you are thinking! The "column space" of a matrix is the space spanned by its columns thought of as vectors. The column space of your matrix is the one dimensional space spanned by <0, 0, 1>= [itex]\vec{k}[/itex]. The null space of a matrix, A, is the set of all vectors, [itex]\vec{v}[/itex] such that [itex]A\vec{v}= \vec{0}[/itex]. For this matrix that is the space spanned by <1, 0, 0>= [itex]\vec{i}[/itex] and <0, 1, 0>= [itex]\vec{j}[/itex]. They are not at all the same. In fact the two are orthogonal complements.

    It is true for any n by n matrix, with n odd, that the null space cannot be the same as the column space because, for any n by n matrix, the sum of the dimension of the column space and the dimension of the null space must equal n. If the two dimensions are the same, their sum is an even number.
     
  4. Aug 31, 2009 #3
    Hi,

    I forgot the "space" in nullspace.

    The book writes:

    n - r = r

    n = 3

    3 = 2r is impossible.

    The n - r = r is confusing me. Is r meant to be the number of pivot columns?

    Thanks!
     
  5. Aug 31, 2009 #4

    You have probably learned a theorem like "rank + nullity = number of columns" or "rank + dimension of null space = number of columns." Yes, rank = number of pivot entries in rre form.

    Also "rank= row rank = col rank = dim of col sp."

    r = dim of col sp
    n-r = dim of null sp

    Set n-r equal to r; is the resulting equation possible?
     
  6. Aug 31, 2009 #5
    Ah yes, n - r is the number of special solutions, number of free variables and the dimension of the nullspace.
    Thank you very much!
     
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