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mt2solns - University of California Los Angeles Midterm...

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University of California, Los Angeles Midterm Examination 2 November 9, 2011 Mathematics 115A Section 5 SOLUTIONS 1. (6 points) Each part is worth 3 points. For each of the following statements, prove or find a counterex- ample. (a) Let V and W be finite dimensional vector spaces and let T : V W be a surjective linear transformation. If β is a basis for V then T ( β ) is a basis for W . Solution: The statement is false. Consider the spaces V = R 2 and W = R 1 with β the standard ordered basis for V . Let T be the unique linear transformation such that T ( e 1 ) = 1 and T ( e 2 ) = 0. The transformation T is surjective since T ( β ) = { 1 , 0 } generates W . However, T ( β ) is not a basis since it is not linearly independent. (b) M m × n ( R ) is isomorphic to P r ( R ) if and only if r = mn - 1. Solution: The statement is true. The finite dimensional real vector spaces M m × n ( R ) and P r ( R ) are isomorphic if and only if they have the same dimension. Since dim(M m × n ( R )) = mn and dim( P r ( R )) = r + 1. The spaces are isomorphic if and only if mn = r + 1, or r = mn - 1.
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