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Unformatted text preview: x + y = 2 2 xy = 1 y = 2 5 NAME: 6. [10 points] Prove that only one of these quadratic forms is positive deﬁnite. q 1 ( x ) = 2 x 2 + 3 y 2 + z 2 + 2 xz + 2 yz q 2 ( x ) = 2 x 2 + 6 y 2 + 2 z 2 + 6 xy + 2 xz + 6 yz 6 NAME: 7. [15 points] The matrix A is the standard matrix representation of a linear operator from R 3 to R 3 . Find a basis in which the matrix representation of the same operator is diagonal. Write down also what the diagonal matrix representation is. Explain your answer . A = 1 0 1 0 4 0 1 0 1 7 NAME: 8. [10 points] Determine whether each of the following is a linear transformation from P (2) to P (1) (Prove your answer. Here p ( x ) is a polynomial in P (2) ) a) L ( p ( x )) = p ( x ) + 2 x b) L ( p ( x )) = p 00 ( x )p ( x ) c) L ( p ( x )) = 2 p ( x )1 8...
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This note was uploaded on 02/10/2009 for the course MATH 415 taught by Professor Bertrandguillou during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 BERTRANDGUILLOU
 Math, Linear Algebra, Algebra

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