Assignment 2 solution

Assignment 2 solution - MATH 223, Linear Algebra Winter...

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MATH 223, Linear Algebra Winter 2008 Solutions to Assignment 2 C stands for the set of complex numbers, and R for the set of real numbers. 1. Here are some subsets of the complex vector space C 3 . In each case, decide whether the given set S is a subspace of C 3 ; justify your answers. (a) S = { a - b 3 ib (2 + i ) a - 2 b : a,b any complex numbers } . Solution: This is a subspace. First ~ 0 S since we can take a = b = 0. Next, suppose that ~v 1 and ~v 2 are in S . Then ~v 1 = a 1 - b 1 3 ib 1 (2 + i ) a 1 - 2 b 2 for some complex numbers a 1 and b 1 , and ~v 2 = a 2 - b 2 3 ib 2 (2 + i ) a 2 - 2 b 2 for some complex a 2 , b 2 . So ~v 1 + ~v 2 = a 1 - b 1 + a 2 - b 2 3 ib 1 + 3 ib 2 (2 + i ) a 1 - 2 b 1 + (2 + i ) a 2 - 2 b 2 = ( a 1 + a 2 ) - ( b 1 + b 2 ) 3 i ( b 1 + b 2 ) (2 + i )( a 1 + a 2 ) - 2( b 1 + b 2 ) , which is of the right form to be an element of S . Hence S is closed under addition. Finally, suppose ~v S and α is a (complex) scalar. Then ~v = a - b 3 ib (2 + i ) a - 2 b for some complex numbers a and b , so α~v = ( αa ) - ( αb ) 3 i ( αb ) (2 + i )( αa ) - 2( αb ) , again an element of S . Thus S is closed under scalar multiplication. ( S is in fact Span { 1 0 2 + i , - 1 3 i - 2 } . ) (b) S = { a - b 3 ib + 2 (2 + i ) a - 2 b : a,b any complex numbers } . Solution: This is not a subspace. In fact, ~ 0 / S . If it were (note the use of the subjunctive) we would need a - b = 3 ib +2 = (2+ i ) a - 2 b = 0 1
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for some complex a and b . The first two equations would tell us a = b = 2 3 i , but then (2 + i ) a - 2 b 6 = 0. So it’s impossible for S to have the zero vector. (In fact, it is not closed under either addition or scalar multiplication, either. But you don’t need to show that now.) (c) S = { a - b 3 ib (2 + i ) a - 2 b : a,b any real numbers } . Solution: This is not a subspace. (Although it does have the zero
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This note was uploaded on 04/29/2008 for the course MATH 223 taught by Professor Loveys during the Fall '07 term at McGill.

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Assignment 2 solution - MATH 223, Linear Algebra Winter...

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