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Assignment 5

# Assignment 5 - 3 Show the exchange property for linear span...

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MATH 223, Linear Algebra Winter, 2008 Assignment 5, due in class Wednesday, February 13, 2008 1. Let V = P 3 ( X ), the space of real polynomials with degree at most 3. Define T : V -→ V by T ( p ( x )) = ( x 2 - 2 x ) p 00 ( x ) + (2 x + 2) p 0 ( x ) - 12 p ( x ) . (a) Verify that T is a linear operator on V . (b) Find [ T ] B where B = (1 , x, x 2 , x 3 ) is the standard ordered basis for V . (c) Find [ T ] B 0 where B 0 = (1 , x +1 , x 3 - x 2 , x 2 ) is a nonstandard ordered basis for V . (d) Find a basis for each of ker ( T ) and im ( T ). (e) Show that T ( T + 12 I )( T + 10 I )( T + 6 I ) = 0. 2. Let V be the vector space M n ( F ) of all n × n matrices with entries from the field F ; let A V be any particular matrix. Show that if we define TX = AX - XA for every X V , then T is a linear operator on V . For the rest of the problem, assume that n = 2, F = R and A = 1 1 1 1 . (a) Let B be the standard ordered basis for V , that is, B = 1 0 0 0 , 0 1 0 0 , 0 0 1 0 , 0 0 0 1 ¶¶ . Find [ T ] B . (b) Let B 0 be the following nonstandard basis for V . B 0 = 1 0 0 1 , 1 1 1 1 , 1 - 1 1 - 1 , 1 1 - 1 - 1 ¶¶ . Find [ T ] B 0 . (c) Find a basis for each of ker ( T ) and im ( T ). 3. Show the exchange property
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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 T-invariant if T ~w ∈ W for all ~w ∈ W . (a) Show that, for any T , { ~ } , V , ker ( T ) and im ( T ) are T-invariant. (b) Show that if W 1 and W 2 are T-invariant, 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 T-invariant subspaces of V . 1...
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