hw10 - -λI ) T = T ( T-λI ) ) 3. Let G be the square in R...

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HW # 10 Due date: 5/3/2011 1. Suppose T 1 and T 2 are linear operators on V such that k T 1 v k = k T 2 v k for all v V Show that dim( rangeT 1 ) = dim( rangeT 2 ) . (This was a step from the Polar Decomposition theorem that we skipped). 2. Show that the operator ( T - λI ) dimV T is equal to the operator T ( T - λI ) dimV . This was a step in the proof of the existence of Jordan Normal form. (hint: first prove ( T
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Unformatted text preview: -λI ) T = T ( T-λI ) ) 3. Let G be the square in R 2 with vertices (0 , 0) , (0 , 1) , (1 , 0), and (1 , 1). Notice G has area 1. Suppose that T ∈ L ( R 2 ) is an linear operator such that the eigenvalues of √ T * T are 4 and 8 with eigenvectors (0 , 1) and (1 , 0). What is the volume of T ( G )? Page 188: 5,6,7 1...
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This note was uploaded on 05/09/2011 for the course MAT 310 taught by Professor Staff during the Spring '08 term at SUNY Stony Brook.

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