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Unformatted text preview: (b} Show that the yz plane is not ^invariant. 7 Let T : V Â» V be a linear operator and let v â‚¬ V satisfy T(v) â€” mv where m is a fixed element of the field F. Now define the subset En = (u e V : T(u) = mil}. (a) Show that E m is a vector subspace of V. (b) Show that E m is Tinvariant. g Let L : V â€” * V be a linear Operator on the ndimcnsioim] vector space V, and let v Â£ V. (a) Let m. be the smallest integer with the property that L m (v) E Span({v,i(v) L m'(v)}. Prove that S. = Span({v, i(v),. .., L m'(v)}). C**""<ir KlÂ»u.) (b) Let L : M 2x2 fft) > A/ 2x2 (K) be given by Find a basis for the /^invariant subspace generated by 1 M oo ( l 001 1 1 1 Definition. Let L : V â€” * V be a linear operator on a vector space V and let v Â£ V. The Lcyclic subspace of V generated by v, S v , is defined by 5, = Span({v, i(v), L 2 (v) ^*( v ), â€¢â€¢ â€¢})...
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 Fall '08
 CELMIN
 Linear Algebra, Derivative, Transformations, Vector Space, linear operator

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