Assignment 5

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ).
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3. Show the exchange property for linear span. That is, suppose that V is a vector space over the eld 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 reection across the line (where goes through the origin). Identify all the T-invariant subspaces of V . 1...
View Full Document

Ask a homework question - tutors are online