Math 113 Homework 5
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چکیده
Proof. Suppose that U1, . . . , Um are invariant subspaces of T ∈ L(V ). We wish to show that U1 + · · ·+ Um is also an invariant subspace. Consider any ~v ∈ U1 + · · ·+ Um. By definition, we can write ~v = ~u1 + · · ·+ ~um, for some ~u1 ∈ U1, . . . , ~um ∈ Um. Since U1, . . . , Um are invariant subspaces, by definition, we know T~u1 ∈ U1 ⊂ U1 + · · ·+ Um, ..., T~um ∈ Um ⊂ U1 + · · ·+ Um. Hence we can conclude, since U1 + · · ·+ Um is a subspace and therefore closed under addition, that T~v = T~u1 + · · ·+ T~um ∈ U1 + · · ·+ Um, i.e. U1 + · · ·+ Um is an invariant subspace.
منابع مشابه
Some notes on sets , logic , and mathematical language
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