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2009amt1web - M346 First Midterm Exam 1 0 0 1a In R3 let E...

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M346 First Midterm Exam, February 11, 2009 1a) In R 3 , let E be the standard basis and let B = 1 2 3 , 0 1 4 , 0 0 1 be an alternate basis. Let v = 3 - 2 14 . Find P EB , P BE and [ v ] B . 1b) In R 2 [ t ], let E = { 1 ,t,t 2 } be the standard basis and let B = { 1+2 t +3 t 2 ,t +4 t 2 ,t 2 } be an alternate basis, and let v = 3 - 2 t +14 t 2 . Find P EB , P BE and [ v ] B . 2. Let L : R 3 R 4 be given by L x 1 x 2 x 3 = x 1 + x 2 + x 3 x 1 + 2 x 2 + 3 x 3 x 1 + 3 x 2 + 5 x 3 x 1 + 4 x 2 + 7 x 3 . a) Find the matrix of L (relative to the standard bases for R 3 and R 4 . b) Let V = { x R 3 : L ( x ) = 0 } . What is the dimension of V ? Find a basis for V . 3. On R 2 , consider the basis B = braceleftbiggparenleftbigg 2 1 parenrightbigg , parenleftbigg 3 2 parenrightbiggbracerightbigg . Let L ( x ) = A x , where A = parenleftbigg 19 - 30 10 - 16 parenrightbigg .
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