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Homework Help: Question about row/column/nullspace

  1. Apr 12, 2017 #1
    1. The problem statement, all variables and given/known data
    If we have a 3x5 matrix:

    The row space is in r5, the col space is in r3, and the nullspace is in r3 correct?
    Because you would need 5 components to be a member of r5 so the col space cannot be a member of r5 correct?

    Here is the question: http://prntscr.com/evo91g

    2. Relevant equations

    3. The attempt at a solution
    Am I getting something fundamentally wrong here? Or is the question just wrong? I believe I'm right for reasons mentioned above.
  2. jcsd
  3. Apr 12, 2017 #2


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    The nullspace is a subspace of ##\mathbb{R}^5##, right? Therefore the nullspace is not in ##\mathbb{R}^3##. For the rest you seem correct. It seems the question is wrong.
  4. Apr 12, 2017 #3
    Why would it be in r5 though? Each vector in the nullspace will only have 3 components
  5. Apr 12, 2017 #4


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    Can you give me your definition of null space?
  6. Apr 12, 2017 #5
    The subspace of linear combinations that make your system equal to 0?

    edit: Now that I think about it, when A is a 3x5 matrix, you need a 5x1 matrix vector X. Since X is 5x1, that means the nullspace is in r5 since X represents all vectors inside the nullspace?
  7. Apr 12, 2017 #6
    The null space and the row space of a matrix will always be sub-spaces of the same vector space (why?).The column space and row space of a matrix will be sub-spaces of the vector space whose dimension is the number of elements in the vector. So if we have a 12x23 matrix, its row space is a sub-space of R23 and its column space is a sub-space of R12. From here I assume you can figure out the correct solution.
  8. Apr 12, 2017 #7
    Yup! Makes sense now. Just find it interesting that I understand the nullspace and can compute it easily but I still abstract details like that.
  9. Apr 13, 2017 #8


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    Yes, your reasoning is correct. That's a very bad definition of null space, though.

    A better definition would be: Let ##A## be an ##m \times n## matrix.

    ##Null(A) := \{x \in \mathbb{R}^n| Ax = 0\}##

    Or if you are familiar with linear mappings: Let ##f: V \rightarrow W## be a linear mapping:

    ##Ker(f) := \{x \in V|f(x) = 0\}##
    Last edited: Apr 13, 2017
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