101st_tut2

101st_tut2 - MATH1111/2010-11/ Tutorial II 1 MATH1111...

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MATH1111/2010-11/ Tutorial II 1 MATH1111 Tutorial II 1. Let A and B be two matrices satisfying AB = ± BA . (a) Must A and B be square matrices of the same order? Explain. (b) Show that if the order is odd, then A or B is singular. (c) Must A or B be singular if the order is even? Justify your answer. 2. Let V be a vector space and x 2 V . A student says that it is ambiguous to write ± 2 x , because one may interpret ± 2 x in di±erent ways, such as ± (2 x ) ; ( ± 1)(2 x ) ; ( ± 2) x ; 2( ± x ) ; (2 ² ( ± 1)) x ; ²²² What is your comment? Give your justi²cation. 3. Are the following sets vector spaces with the indicated operations? If not, why not? (a) The set V of all polynomials of exact degree 3 together with 0; operations are the usual operations on polynomials. (b) The set V of 2 ³ 2 matrices whose determinant equal zero; operations are usual matrix operations. (c) The set V of 2 ³ 2 matrices whose column sums are equal; operations are usual matrix operations.
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101st_tut2 - MATH1111/2010-11/ Tutorial II 1 MATH1111...

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