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Math Help - column space, null space

  1. #1
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    column space, null space

    Here are my problems:

    Prove that if C is am mxn matrix over the field R of reals then R^n-Col(C^T) (direct sum) N(C), where Col (A) denotes the column space of the matrix A and N(A) denotes the right null space of A.

    Plese, help!
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  2. #2
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    Quote Originally Posted by mivanova View Post
    Prove that if C is am mxn matrix over the field R of reals then R^n-Col(C^T) (direct sum) N(C), where Col (A) denotes the column space of the matrix A and N(A) denotes the right null space of A.
    Note sure what the question is asking? Is that that this expression is zero?
    If thus, I think you need to use the rank nullity theorem.
    Define T(\bold{x}) = A\bold{x} as a linear transformation A: \mathbb{R}^n \to \mathbb{R}^m.
    But \text{rank}(A) is the dimension of the column space.
    And \text{nullity}(A) is the dimension of the null space.
    Thus the dimensions add up to n, the dimension of \mathbb{R}^n.
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  3. #3
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    column space, null space

    Hi,
    Thank you so much for your answer. I don't think that this expressiin is zero. The way that is giving to us is to prove that R^n=Col(C^T)(dir sum)N(C).
    And, sorry if it is a stupid questions, but what differrence will it make if it is "right" null space or "left"?

    Thank you so much!
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