Unformatted text preview: : If L : V → W is linear, and T is a subspace of W , then L1 ( T ) is a subspace of V . b) Prove that dim(Row( A )) = dim(Col( A )). [If you follow my outline, explain each step and include the phrase ”dependency relation” when it is needed]. c) Show that if A is similar to B then A T is similar to B T . Answers: 1) FTTTF 2) [6 7] T . I used [3 4] T = 2[2 3] T[1 2] T to get this, but there are other ways (you could ﬁnd the matrix rep for L instead, for example). 3) a) = HW and b) = Thm. The easiest is part c): Since A = S1 BS , we can transpose both sides and get A T = S T B T ( S1 ) T . But ( S1 ) T = ( S T )1 , by previous HW, so A T is similar to B T . 1...
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Matrices, MxN matrix, row equivalent matrices, Prof. S. Hudson

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