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homework_3 - THE CHINESE UNIVERSITY OF HONG KONG Department...

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications (Fall 2007) Homework 3 Name: Student No.: Class: Final Result: I acknowledge that I am aware of University policy and regulations on hon- esty in academic work, and of the disciplinary guidelines and procedures appli- cable to breaches of such policy and regulations, as contained in the website http://www.cuhk.edu.hk/policy/academichonesty/ Signature Date Choose 7 out of the the following 10 problems; Questions (1), (2), (3), (4), (5) and (6) must be answered. 1. In Problems (a) and (b), a set of objects is given together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not, list all axioms that fail to hold. (a) The set of all positive real numbers x with the operations x + x 0 = xx 0 and k x = x k . (b) The set of all 2 × 2 matrices of the form a 0 0 b with matrix addition and scalar multiplication.
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