Unformatted text preview: 3. Show the exchange property for linear span. That is, suppose that V is a vector space over the ﬁeld F and S ∪ { ~v, ~w } is a subset of F ; also suppose that ~w ∈ span ( S ∪ { ~v } ) but ~w / ∈ span ( S ). Show that (in this case) ~v ∈ span ( S ∪ { ~w } ). 4. Suppose that T is a linear operator on the vector space V , and that W is subspace of V . We say that W is Tinvariant if T ~w ∈ W for all ~w ∈ W . (a) Show that, for any T , { ~ } , V , ker ( T ) and im ( T ) are Tinvariant. (b) Show that if W 1 and W 2 are Tinvariant, then so are W 1 ∩ W 2 and W 1 + W 2 . (c) Suppose that V = R 2 and T = T ‘ is the reﬂection across the line ‘ (where ‘ goes through the origin). Identify all the Tinvariant subspaces of V . 1...
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 Fall '07
 Loveys
 Linear Algebra, Algebra, Polynomials, Vector Space, linear operator, nonstandard ordered basis

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