s11_mthsc853_hw04

# s11_mthsc853_hw04 - A-1(b Show that if either A or B is...

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MthSc 853 (Linear Algebra) Dr. Macauley HW 4 Due Monday, February 14th, 2011 (1) Show that whenever meaningful, (i) ( ST ) 0 = T 0 S 0 (ii) ( T + R ) 0 = T 0 + R 0 (iii) ( T - 1 ) 0 = ( T 0 ) - 1 . Here, S 0 denotes the transpose of S . Carefully describe what you mean by “whenever meaningful” in each case. (2) Give a direct algebraic proof of N T 0 = R T (i.e., don’t just use the fact that N T 0 = R T and take the annihilator of both sides.) (3) Let X be a ﬁnite-dimensional vector space and A,B L ( X,X ) (linear functions X X ). (a) Show that if A is invertible and similar to B , then B is also invertible, and B - 1 is similar to
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Unformatted text preview: A-1 . (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 I-T 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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