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Math Help - Linear Transformations: im(S+T) subset of im(S) + im(T)

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    Linear Transformations: im(S+T) subset of im(S) + im(T)

    Let V be an n-dimensional vector space over R, and let S and T be linear transformations from V to V.

    (i) Show that im(S+T) is a subset of im(S) + im(T)
    (ii) Show that r(ST) <= min(r(S),r(T)), and that n(ST) <= n(S) + n(T)
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by csmajor9 View Post
    Let V be an n-dimensional vector space over R, and let S and T be linear transformations from V to V.

    (i) Show that im(S+T) is a subset of im(S) + im(T)
    (ii) Show that r(ST) <= min(r(S),r(T)), and that n(ST) <= n(S) + n(T)
    I'll help with i) and the first part of ii) and leave the rest to you. For i) merely note that if v\in\text{im}\left(S+T\right) then v=(S+T)(w) for some w\in V. But, (S+T)(w)=S(w)+T(w)\in \text{im}(S)+\text{im}(T)...conclude.

    For the first part of ii) the fact that \text{rk}(ST)\leqslant \text{rk}(S) is clear since \text{im}(ST)=(ST)(V)=S(T(V))\subseteq S(V)=\text{im}(S). Now, to see that \text{rk}(ST)\leqslant \text{rk}(T) note that in general the image of a linear homomorphism has lesser dimension than the domain etc.
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