Unformatted text preview: A1 . (b) Show that if either A or B is invertible, then AB and BA are similar. (4) Suppose T : X → X is a linear map of rank 1, and dim X < ∞ . (a) Show that there exists a unique c ∈ K such that T 2 = cT . (b) Show that if c 6 = 1, then IT has an inverse. (5) Suppose that S,T : X → X and dim X < ∞ . (a) Show that rank( S + T ) ≤ rank( S ) + rank( T ). (b) Show that rank( ST ) ≤ rank( S ). (c) Show that dim( N ST ) ≤ dim N S + dim N T ....
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 Spring '08
 Staff
 Linear Algebra, Algebra, finitedimensional vector space, dim nt, Dr. Macauley HW, dim NS

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