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Proof regarding Range of linear maps

  1. Feb 2, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that if T, S : V -> W are linear and W is finite dimensional then
    dim (range(S + T)) <= dim range S + dim range T

    2. Relevant equations

    3. The attempt at a solution

    I know that the dim range S + dim range T can be at most 2 * dim W. However, I'm a little stuck on trying to figure out what dim (range(S+T)) is. How do I determine S + T?
  2. jcsd
  3. Feb 2, 2012 #2


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    Staff Emeritus
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    Suppose Range S and Range T have only the 0 vector in common. What is S+ T? If they have more than the 0 vector in common, their intersection must be a subspace.

    Or is the problem that you do not know how S+ T is defined? It is the space of all vectors orf the form u+ v where u is in S and v is in T.
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