Math121sol6 - Math 121: Linear Algebra and Applications...

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Unformatted text preview: Math 121: Linear Algebra and Applications Prof. Lydia Bieri Solution Set 6 Posted: November 30th Written by: Luca Candelori Exercise 1 (2.6/1d-h, 2.7/1c-g) . (a) (d) TRUE. Note that for any finite dimensional V , V = ( V * ) * (e) FALSE. Let V = R 3 and consider the standard basis = { e 1 ,e 2 ,e 3 } . Its dual basis in ( R 3 ) * is given by { g 1 ,g 2 ,g 3 } where g 1 ( x,y,z ) = x,g 2 ( x,y,z ) = y,g 3 ( x,y,z ) = z . Take now the basis S = { f 1 ,f 2 ,f 3 } given in Exercise 2.6.4 from pset 5 and consider the map T ( e i ) = f i for i = 1 , 2 , 3. This is an isomorphism, but T ( ) = S 6 = * . (f) TRUE. By definition (g) TRUE. For any finite dimensional V , V V * . Therefore V * V W W * i.e. V * W * . (h) FALSE. The derivative of a function is a linear function on the vector space of all differentiable functions, but it is not a functional unless this space has dimension 1. (b) (c) FALSE. The solutions of the auxiliary polynomial yield solutions to the equation, but the polynomial itself is not a solution. (d) FALSE. Its a linear combination of those. (e) TRUE. This is true for linear equations, by a simple check. (f) FALSE. You also need to consider the multiplicity of the zeroes. (g) TRUE. Just substitute the expression y ( i ) to t i for each i . Exercise 2 (2.6/9) ....
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Math121sol6 - Math 121: Linear Algebra and Applications...

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