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Unformatted text preview: othogonal linear forms. λ = −3, 1
− x2 − 4xy − y2 = A= xy −1 −2
−2 −1 =− −1 −2
−2 −1 x
y 3
2 11
11
=− + 1
2 1 −1
−1
1 3
1
( x + y) 2 +
( x − y) 2
2
2 [4] Express the quadratic form
2x2 − 4xy + 5y2
as a sum of squares of othogonal linear forms. λ = 1, 6 2x2 − 4xy + 5y2 = A= xy 2 −2
−2
5
2 −2
−2
5 = 1
5 42
21 x
y = + 6
5 1 −2
−2
4 1
6
(2x + y)2 +
(x − 2y)2
5
5 , 2 × 2 Exercise Set I (quadratic forms), November
[5] Express the quadratic form
2x2 + 4xy − y2
as a sum of squares of othogonal linear forms. λ = −2, 3 2
2
2 −1 A= 2x2 + 4xy − y2 = 2
2
2 −1 xy 2
5 =−
x
y 1 −2
−2
4 =− + 3
5 42
21 3
2
(x − 2y)2 +
(2x + y)2
5
5 [6] Express the quadratic form
3x2 + 2xy + 3y2
as a sum of squares of othogonal linear forms. λ = 2, 4 31
13 A= 3x2 + 2xy + 3y2 = 1 −1
−1
1 = 31
13 xy x
y 11
11 +2 = ( x − y) 2 + 2 ( x + y ) 2 [7] Express the quadratic form
− 2x2 + 4xy + y2 as a sum of squares of othogonal linear forms. λ = −3, 2 A= − 2x2 + 4xy + y2 = xy −2 2
21 =− −2 2
21 3
5 x
y 4 −2
−2
1
=− + 2
5 12
24 2
3
( 2x − y ) 2 +
( x + 2y ) 2
5
5 [8] Express the quadratic form
− x2 − 8xy − y2 as a sum of squares of othogonal linear forms. λ = −5, 3
− x2 − 8xy − y2 = A= xy −1 −4
−4 −1 =− −1 −4
−4 −1 x
y 5
2 11
11
=− + 3
2 1 −1
−1
1 5
3
( x + y) 2 +
( x − y) 2
2
2 , 2 × 2 Exercise Set J (recurrence relations), November 2 × 2 Exercise Set J (recurrence relat...
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 Spring '14
 DaveBayer
 Linear Algebra, Algebra

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