2009bsol1 - M346 First Midterm Exam Solutions, September...

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Unformatted text preview: M346 First Midterm Exam Solutions, 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 . P EB = ([ b 1 ] E [ bb 2 ] E ) = parenleftbigg 2 5 3 7 parenrightbigg . P BE = P- 1 EB = parenleftbigg- 7 5 3- 2 parenrightbigg . Here we used the fact that parenleftbigg a b c d parenrightbigg- 1 = 1 ad- bc parenleftbigg d- b- c a parenrightbigg . b) If v = parenleftbigg 4 1 parenrightbigg , find [ v ] B . Since [ v ] E = parenleftbigg 4 1 parenrightbigg , [ v ] B = P BE [ v ] E = parenleftbigg- 23 10 parenrightbigg . c) Solve the system of equations: 2 x 1 + 5 x 2 = 4; 3 x 1 + 7 x 2 = 1. This is the exact same problem as (b), namely writing parenleftbigg 4 1 parenrightbigg as a linear combination of parenleftbigg 2 3 parenrightbigg and parenleftbigg 5 7 parenrightbigg . The solution, as before, is x 1 =- 23 , x 2 = 10 ....
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2009bsol1 - M346 First Midterm Exam Solutions, September...

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