5.2(2d,3b,7,8,12) - Linear Algebra -115 Solutions to...

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Unformatted text preview: Linear Algebra -115 Solutions to Eleventh Homework Problem 2( d ) (Section 5 . 2) det ( A- λI ) = 7- λ- 4 8- 5- λ 6- 6 3- λ = (3- λ ) 7- λ- 4 8- 5- λ = (3- λ )[(7- λ )(- 5- λ ) + 32] = (3- λ )( λ 2- 2 λ- 3) = (3- λ )( λ- 3)( λ + 1) . So,- 1 , 3 , 3 are the eigenvalues. We need to find the eigenspaces, too. A + 1 I = 8- 4 8- 4 6- 6 4 ∼ 2- 1 3- 3 2 . So, ( x, y, z ) ∈ E- 1 iff 2- 1 3- 3 2 x y z = iff 2 x- y = 0 and 3 x- 3 y + 2 z = 0 iff y = 2 x and- 3 x + 2 z = 0 iff y = 2 x and z = 3 2 x, i.e. E- 1 = { x (2 , 4 , 3) } = span (2 , 4 , 3). A- 3 I = 4- 4 8- 8 6- 6 ∼ 4- 4 . So, ( x, y, z ) ∈ E 3 iff 4- 4 x y z = iff 4 x- 4 y = 0 iff x = y i.e. E 3 = { x (1 , 1 , 0) + z (0 , , 1) } = span { (1 , 1 , 0) , (0 , , 1) } ....
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This note was uploaded on 02/02/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

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5.2(2d,3b,7,8,12) - Linear Algebra -115 Solutions to...

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