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# homework_4 - 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 4 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 Answer all the questions. 1. Suppose the n × n matrix A has rank n - r . Let x 1 , x 2 , · · · , x r be linearly independent vectors satisfying A x = 0 . Suppose x is a vector and A x = 0 . Explain why x must be a linear combination of x 1 , x 2 , · · · , x r . 2. For each of the following matrices, find a basis for the row space, a basis for the column space, and a basis for the nullspace. (a) A = 1 - 1 3 5 - 4 - 4 7 - 6 2 (b) A = 1 4 5 2 2 1 3 0 - 1 3 2 2 . 3. Find the rank and nullity of the matrix; the verify that the values obtained satisfy rank A + nullity A = n , where n

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homework_4 - THE CHINESE UNIVERSITY OF HONG KONG Department...

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