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mterm2 211 - orthogonal basis in the image of T c(10pt...

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Stony Brook University Introduction to Linear Algebra Mathematics Department MAT 211 Julia Viro April 10, 2008 Midterm II Examination time: 5:20-6:40 pm. No electronic devices, books or notes. Show all your work. Name Student ID # Yes, I give my permission to put my grades on the web. Problem # Points/total 1 /30 2 /30 3 /30 4 /10 Total /100
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2 Name Problem 1 (30pt). A linear transformation T : R 3 R 2 is defined by T ( x, y, z ) = ( x + 2 y + 3 z, 4 x + 5 y + 6 z ) . Find the matrix of T with respect to the standard basis in R 3 and the basis B = { (1 , 2) , (1 , 3) } in R 2 .
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3 Name Problem 2 (30pt). Let P 2 be the vector space of polynomials of degree 2 equipped with the inner product < p, q > = 1 - 1 p ( x ) q ( x ) dx. Define a linear transformation T : P 2 → P 2 by the formula T ( p ) = (1 - x 2 ) p - 2 xp + 6 p. a) (10pt) Find the matrix of T with respect to the basis B = { 1 , x, 3 x 2 - 1 } of P 2 . b) (10pt) Find an orthogonal basis in the kernel and an
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Unformatted text preview: orthogonal basis in the image of T . c) (10pt) Find (Ker T ) ⊥ and (Im T ) ⊥ . 4 Name 5 Name Problem 3 (30pt). The former secret agent is now a constructor. His company works in the Euclidean space R 3 . He has got a tool called the Gram-Schmidt orthogonalization and a basis { (1 , 1 , 1) , (2 , , 1) , (2 , 4 , 5) } of R 3 . Help him to construct an orthonormal basis of R 3 out of the given basis! 6 Name 7 Name Problem 4 (10pt). Let V be a finite dimensional inner product space. a) (3pt) Formulate the Cauchy-Schwarz inequality b) (2pt) Formulate the Triangle inequality c) (2pt) Formulate the Pythagorean theorem d) (3pt) Which transformation of V is called orthogonal?...
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