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Engineering Calculus Notes 403

Engineering Calculus Notes 403 - 1 −→ u 1 ξ 2 −→ u...

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3.9. THE PRINCIPAL AXIS THEOREM 391 (a) f ( x,y,z ) = 5 x 2 + 3 y 2 + z 2 2 xy + 2 yz 6 x 8 y 2 z (b) f ( x,y,z ) = x 2 + y 2 + z 2 + xy + yz + xz 2 x (c) f ( x,y,z ) = x 2 + y 2 + z 2 + xy + yz + xz 3 y z (d) f ( x,y,z ) = x 2 + y 2 + z 2 + xy + yz + xz 2 x 3 y z (e) f ( x,y,z ) = 2 x 2 + 5 y 2 6 xy + 2 xz 4 yz 2 x 2 z (f) f ( x,y ) = x 3 + x 2 3 x + y 2 + z 2 2 xz (g) f ( x,y ) = x 3 + 2 x 2 12 x + y 2 + z 2 2 xy 2 xz Theory problems: 3. (a) Adapt the proof of Proposition 3.9.2 to show that if M = p a b b c P is a symmetric 2 × 2 matrix, then there exist two unit vectors −→ u 1 and −→ u 2 and two scalars λ 1 and λ 2 satisfying M −→ u i = λ i −→ u i for i = 1 , 2 . (b) Show that if −→ u i , i = 1 , 2 , 3 are orthonormal vectors, then an arbitrary vector −→ v R 3 can be expressed as −→ x = ξ
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Unformatted text preview: 1 −→ u 1 + ξ 2 −→ u 2 where ξ i = −→ x · −→ u i for i = 1 , 2 . (c) Show that if M = [ Q ] and −→ x = ( x,y ) then Q has the weighted squares decomposition Q ( x,y ) = λ 1 ξ 2 1 + λ 2 ξ 2 2 . 4. Let A = p a b c d P be any 2 × 2 matrix. (a) Show that the characteristic polynomial det( A − λI ) is a polynomial of degree 2 in the variable λ . (b) Show that if A is 3 × 3, the same polynomial is of degree 3....
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