math mock final

# math mock final - MATH 54 mock nal test All the necessary...

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MATH 54, mock final test. All the necessary work to justify an answer and all the necessary steps of a proof must be shown clearly to obtain full credit. Partial credit may be given but only for significant progress towards a solution. Show all relevant work in logical sequence and indicate all answers clearly. Cross out all work you do not wish considered. 2 pages of notes are allowed. Books and electronic devices are not allowed during the test. 1. Determine for which values of b 1 , b 2 , b 3 the system 2 x 1 - 4 x 2 - 2 x 3 = b 1 - 5 x 1 + x 2 + x 3 = b 2 7 x 1 - 5 x 2 - 3 x 3 = b 3 has a solution. 2. Let T : IP 2 IR 3 be the linear map taking a polynomial of degree at most 2 to its values at the points - 1, 0 and 1: T ( p ) := p ( - 1) p (0) p (1) . (a) Write the matrix of T in the standard bases. (b) Using the result of (a), find a polynomial f IP 2 such that f ( - 1) = 1, f (0) = 0, f (1) = 2. 3. Let S be the linear space of functions of the form S := { p ( x ) e 2 x : p IP 2 } and L be the differential operator L :=( D - 2 I ) 2 = ( D - 2 I )( D - 2 I ) where I

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