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082nd_tut2sol - MATH1111/2008-09/Tutorial II Solution 1...

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MATH1111/2008-09/Tutorial II Solution 1 Tutorial II Suggested Solution 1. The purpose of this question is to understand the two quantifiers: for any (or for all ) and there exists (or for some ). I. Decide, based on common sense, which of the following is true. (i) Cantonese is the mother tongue for all HKU students. [This statement can be rephrased as: Cantonese is the mother tongue for any HKU student.] (ii) English is the mother tongue for some HKU students. [This statement can be rephrased as: There exists a HKU student whose mother tongue is English.] (iii) For any child, there exists a woman who is the mother of the child. (iv) There exists a woman such that for any child, she is the mother of the child. II. Prove or disprove the following: (a) For any non-singular matrix A , there exists a matrix B of the same order as A such that A 2 B = A . (b) There exists a matrix B such that for any non-singular matrix A of the same order as B , A 2 B = A . (c) For any non-singular matrix A of order n , there exists a vector x R n such that for any vector y R n , A x = y . (d) For any non-singular matrix A of order n , for any vector y R n , there exists a vector x R n such that A x = y .
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