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# midterm 2 practice2 answers - Math 311 second practice exam...

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Math 311 second practice exam answers 1. ( D 2 - I ) u ( x ) = u ( x ) - u ( x ) = - 4 cos 2 x - cos 2 x = - 5 cos 2 x. 2. To show that these polynomials span the whole vector space, we note that for any a 0 , a 1 , a 2 , a 0 + a 1 x + a 2 x 2 = a 0 1 + a 1 ( x + x 2 ) + ( a 2 - a 1 ) x 2 . To show that these polynomials are linearly independent, suppose that for some a, b, c , a 1 + b ( x + x 2 ) + cx 2 = 0 . Then a + bx + ( b + c ) x 2 = 0 , from which is follows that a = 0 , b = 0 , b + c = 0 , so also c = 0 . 3. Row reduction brings the matrix 1 1 - 1 2 4 2 3 5 1 to the form 1 1 - 1 0 1 2 0 0 0 . The first two columns are lead columns, so the first two of the original vectors Span 1 2 3 , 1 4 5 form a basis for this subspace. In particular, that subspace has dimension 2 . 4. 2 - 2 1 0 1 3 0 0 1 . Since the matrix is upper-triangular, we see right away that the eigenvalues are λ = 1 , 2 . For λ = 1 , the eigenvectors are elements of the null space Null 1 - 2 1 0 0 3 0 0 0 = Null 1 - 2 0 0 0 1 0 0 0 = Span 2 1 0 .

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midterm 2 practice2 answers - Math 311 second practice exam...

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