204ps42006 - Economics 204 August 2006 Due Tuesday 15...

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Problem Set 4 Economics 204 - August 2006 Due Tuesday, 15 August in Lecture Revised 8/13/06, see ** in Problem 4 1. Let Θ be the set of all continuous functions whose domain is the unit interval [0 , 1] and range is R . Let Φ be the subset consisting of all real polynomials (whose domain is restricted to the unit interval) of degree at most two: Φ ≡ { a + bx + cx 2 | a, b, c R } Note that the set Θ is a vector space over the field of real numbers and the subset Φ is a proper subspace. (a) Are the vectors { x, ( x 2 1) , ( x 2 + 2 x + 1) } linearly independent over R ? (b) Find a Hamel basis for the subspace Φ. (c) What is the dimension of Φ ? What is the dimension of Θ ? 2. Recall that a reflection across the x -axis can be achieved with the transformation ( x, y ) ( x, y ). Derive a transformation, T , which reflects a point across the line y = 3 x . (a) First, calculate the action of T on the points (1 , 3) and ( 3 , 1). (b) Next, write the matrix representation of T using these two vectors as bases.
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