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# ex 1 - Find a matrix representation for the operator L of...

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Exercises for Applied Linear Algebra - Math 270 February 24, 2011 Notation: Let P n be the space of real polynomials of degree less or equal than n , e.g.: P 1 = { ax + b, for a, b R } , P 2 = { ax 2 + bx + c, for a, b, c R } . Problem 1 Find two different PLU decompositions for A = 2 - 4 2 - 7 . Problem 2 Show that the space P 2 is spanned by the set 1 , 2 + 2 t, 1 - t + t 2 , 2 - t 2 . Problem 3 Let B = 1 + t + t 2 , 1 + t, 1 be a basis for P 2 . Find the coordinates of 2 - 3 t + 6 t 2 with respect to B . Problem 4 Find an orthonormal basis for U = span [1 0 1] T , [2 1 4] T R 3 . Write the 3 × 3 matrix which represents the orthogonal projection onto U . Problem 5 Let L : P 2 P 1 be the linear application defined by Lp ( x ) = p 00 ( x ) - p 0 ( x ). Find Ker( L ), Im( L ), and the general solution of the equation Lp ( x ) = 2 - 2 x.
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Unformatted text preview: Find a matrix representation for the operator L of the previous exercise. Solve the same exercise using only linear algebra. Problem 7 Find the matrix of the linear transformation L : R 2 → R 2 such that L ([1 ,-√ 3] T ) = [0 , 0] T and L ([ √ 3 , 1] T ) = [3 , √ 3] T . Problem 8 Find the solutions of the equation z 3 = 1 + i . Problem 9 Find the real Fourier coeﬃcients a ,a 1 ,a 2 of the functions f,g,h deﬁned by f ( x ) = µ-1 if x ∈ (-π, 0] 1 if x ∈ (0 ,π ] ; g ( x ) = cos(23 x ); h ( x ) = µ if x ∈ (-π, 0] 1 if x ∈ (0 ,π ] ....
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