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Math Help - Restriction on a linear operator

  1. #1
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    Restriction on a linear operator

    Prove that the restriction of a linear operator T to a T-invariant subspace is a linear operator on that subspace.

    For this proof, I'm not sure if I fully understand the problem.

    Now, let W be a vector space, then T(W) is a subset of W. So this is the restriction. So I have to show that as a linear operator? How?
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hello

    The subspace considered is not necessarily the whole space W, it is a subspace which is T-invariant. If we call it V\subset W, it means that for all x\in V,\,T(x) \in V. Now you've to show that the restriction of T to V (let's call it T_v) is a linear operator, that is to say :
    •  \forall x,\,y\in V,\,T_v(x)+T_v(y)=T_v(x+y)
    • \forall \lambda\in\mathbb{K},\,x\in V,\,T_v(\lambda x)=\lambda T_v(x)

    Aknoledegded that T is a linear operator on W, it should not be a problem.
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