M 346 - First Midterm Fall 2009 - Test

M 346 - First Midterm Fall 2009 - Test - M346 First Midterm...

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Unformatted text preview: M346 First Midterm Exam, September 18, 2009 1) In R 2 , let E = braceleftbiggparenleftbigg 1 parenrightbiggparenleftbigg 1 parenrightbiggbracerightbigg be the standard basis and let B = braceleftbiggparenleftbigg 2 3 parenrightbigg , parenleftbigg 5 7 parenrightbiggbracerightbigg be an alternate basis. a) Find P EB and P BE . b) If v = parenleftbigg 4 1 parenrightbigg , find [ v ] B . c) Solve the system of equations: 2 x 1 + 5 x 2 = 4; 3 x 1 + 7 x 2 = 1. 2. In R 1 [ t ], let E = { 1 , t } be the standard basis and let B = { 4 + 5 t, 3 + 4 t } be an alternate basis. Let L : R 1 [ t ] R 1 [ t ] be the linear transformation L ( a + a 1 t ) = (16 a- 12 a 1 ) + (20 a- 15 a 1 ) t . a) Find P EB and P BE . b) Find the matrix of L relative to the standard basis (that is, find [ L ] E ). c) Find the matrix of L relative to the basis B (that is, find [ L ] B ). [The answer to (c) is much simpler than the answer to (b) and illustrates why we use bases like B .] 3. Let A =...
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M 346 - First Midterm Fall 2009 - Test - M346 First Midterm...

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