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A question about the rank of the sum of linear transformations

  1. Apr 12, 2009 #1
    Notations:
    L(V,W) stands for a linear transformation vector space form vector space V to W.
    rk(?) stands for the rank of "?".

    Question:
    Let τ,σ ∈L(V,W) , show that rk(τ + σ) ≤ rk(τ) + rk(σ).
    I want to know wether the way I'm thinking is right or not, or there's a better explanation.
    My thought is:
    Since every linear tansformation is reprensted by a matrix, also is τ + σ. rk(τ) and rk(σ) roughly means the numbers of basis vectors of their own matrix column spaces, so the combination τ + σ only preserves the distinct basis vectors from the column spaces of the matirces of τ and σ, namely, rk(τ + σ) ≤ rk(τ) + rk(σ) holds.

    Thanks for any help.
     
  2. jcsd
  3. Apr 15, 2009 #2
    Show that the im(τ + σ) is a subspace of im(τ) + im(σ). Remember that in general, rk(τ) = dim(im(τ)).

    (By im, I mean image.)
     
  4. Apr 16, 2009 #3
    Indeed, it's a simpler way to solve the question.

    Thanks!
     
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