2.5+4.4

# 2.5+4.4 - i-2-4 i ± ± ± ± ± ± R 1-3 R 2 =-1-i ± ±...

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Linear Algebra -115 Solutions to Ninth Homework Problem 6( b ) (Section 2 . 5) Since L A is deﬁned to be L A ( x ) = Ax , we have that [ L A ] S = A , where S is the standard basis. We know that [ L A ] β = [ Id ] β S [ L A ] S [ Id ] S β = [ Id ] β S A [ Id ] S β = Q - 1 AQ. So, this Q we are looking for is [ Id ] S β . Since we know β = { ± 1 1 ² , ± 1 - 1 ² } , then [ Id ] S β is just the matrix ± 1 1 1 - 1 ² . ± Problem 2( d ) (Section 4 . 4) ³ ³ ³ ³ ± 3 4 i - 6 i 2 i ²³ ³ ³ ³ = 6 i + 24 i 2 = 6 i - 24 . ± Problem 3( f ) (Section 4 . 4)Expanding det ( A )along the third column we get A = ³ ³ ³ ³ ³ ³ i 2 + i 0 - 1 3 2 i 0 - 1 1 - i ³ ³ ³ ³ ³ ³ = 0 - 2 i ³ ³ ³ ³ ± i 2 + i 0 - 1 ²³ ³ ³ ³ + (1 - i ) ³ ³ ³ ³ ± i 2 + i - 1 3 ²³ ³ ³ ³ = - 2 i ( - i ) + (1 - i )(3 i + 2 + i ) = - 2 + 4 i + 2 - 2 i + 4 = 4 + 2 i. ± 1

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Problem 4( f ) (Section 4 . 4) ± ± ± ± ± ± - 1 2 + i 3 1 - i i 1 3 i 2 - 1 + i ± ± ± ± ± ± ( - 1 - i ) R 3 = ( - 1 - i ) ± ± ± ± ± ± - 1 2 + i 3 1 - i i 1 3 - 3 i - 2 - 2 i 2 ± ± ± ± ± ± R 3 - 2 * R 2 = ( - 1 - i ) ± ± ± ± ± ± - 1 2 + i 3 1 - i i 1 1 -
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Unformatted text preview: i-2-4 i ± ± ± ± ± ± R 1-3 * R 2 = (-1-i ) ± ± ± ± ± ± -4 + 3 i 2-2 i 1-i i 1 1-i-2-4 i ± ± ± ± ± ± = (-1-i )(-1) ± ± ± ± ²-4 + 3 i 2-2 i 1-i-2-4 i ³± ± ± ± = (-1-i )(-1)[(-4 + 3 i )(-2-4 i )-(1-i )(2-2 i )] = (1 + i )[(8 + 12 + 16 i-6 i )-(2-2-2 i-2 i )] = (1 + i )(20 + 14 i ) = 20 + 14 i + 20 i-14 = 6 + 34 i ± Note: I hope that the calculations are correct. If you ﬁnd any mistakes, please, let me know. Thanks. 2...
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## This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.

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2.5+4.4 - i-2-4 i ± ± ± ± ± ± R 1-3 R 2 =-1-i ± ±...

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