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solns1

# solns1 - Mathematics 105 — Spring 2004 — M Christ 1...

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Unformatted text preview: Mathematics 105 — Spring 2004 — M. Christ 1 Solutions to selecta from problem set #1 No number . Let T : R n → R m be a linear transformation. Show that there exists some finite constant M such that | T ( v ) | ≤ M | v | for every v ∈ R n . (This was not assigned, but I used it in class and promised to provide a proof.) Solution. Let { e 1 , ··· , e n } denote the standard basis for R n . Define the finite number B = max( | T ( e 1 ) | , ··· , | T ( e n ) | ). Consider any v ∈ R n and write v = ( v 1 , ··· , v n ) = ∑ n j =1 v j e j . Then | T ( v ) | = | n X j =1 v j T ( e j ) | ≤ n X j =1 | v j | · | T ( e j ) | ≤ B n X j =1 | v j | . Now | v j | ≤ | v | for all j , so we find that | T ( v ) | ≤ Bn | v | . Thus it suffices to define M = Bn . (Actually B √ n would work, but there’s no bonus for efficiency here.) I-4 . Prove that for any x, y ∈ R n , | x | - | y | ≤ | x- y | . Solution. First of all, if you tried to write this out using square roots and so on, you probably encountered a bit of a mess. A better way to do this is to remember that absolute values are often best reasoned about by cases, and to note that on the left-hand side of the inequality, the outermost symbols denote the absolute value of the real number | x | - | y | . So split the proof into two cases. If | x | ≥ | y | then we’re asked to show that | x | - | y | ≤ | x- y | . This is equivalent to | x | ≤ | y | + | x- y | . Since | x | = | y + ( x- y ) | , this is a direct consequence of the triangle inequality. The problem is symmetric, in the sense that if the roles of x, y are interchanged, then neither side of the inequality changes at all. Thus the case where | y | ≥ | x | is exactly the same as the case where | x | ≥ | y | . I-14 . Prove that the union of any family of open sets is open. Prove that the inter- section of any finite family of open sets is open, and give a counterexample showing that this need not hold for infinite families....
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solns1 - Mathematics 105 — Spring 2004 — M Christ 1...

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