10aAss3 - AS3/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG...

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AS3/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 3 Due date : Nov 9, 2010 before 6:30 p.m. Where to hand-in : Assignment Box outside the lifts on the 4th ±oor of Run Run Shaw Remember : Plagiarism is a misconduct Part I: Hand-in your solutions 1. Let V be a vector space, and B = [ b 1 ; ±±± ;b n ] be an ordered basis for V . (So dim V = n .) (a) Show that u 1 ; ±±± ;u r 2 V are linearly independent if and only if [ u 1 ] B ; ±±± ; [ u r ] B are linearly independent vectors in R n . (b) Given any vectors u 1 ; ±±± ;u m 2 V and w 2 V , show that w is a linear combination of u 1 ; ±±± ;u m if and only if [ w ] B is a linear combination of the vectors [ u 1 ] B ; ±±± ; [ u m ] B in R n . [Think about what’s the use of the result in Qn 1.] 2. Let P 3 be the vector space of all polynomials of degree ² 2. De²ne p 1 ( t ) = 1 + t 2 ; p 2 ( t ) = 2 ³ t + 3 t 2 ; p 3 ( t ) = 1 + 2 t ³ 4 t 2 : (a) Use coordinate vectors to show that these polynomials form a basis for P 3 .
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This note was uploaded on 05/04/2011 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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10aAss3 - AS3/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG...

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