04spex2sam - x pp ( t )-4 x p ( t ) + 3 x ( t ) = cos t, x...

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Math 320 Sample Second In-class Exam April 13, 2004 M´arton Bal´azs 1. Decide if the sets S 1 and S 2 below are vector spaces or not. Show a basis for whichever of them is a vector space. (a) (15 points) S 1 is the set of A n × n matrices for which A T = A - 1 . (b) (15 points) S 2 is the set of B 2 × 2 matrices for which B · p 1 - 1 P = p 0 0 P . 2. (40 points) By computing the Wronskian, decide if sin x, cos x, tan x are linearly independent functions on the real interval ( 0 , π 2 ) . 3. (50 points) Solve the initial value problem
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Unformatted text preview: x pp ( t )-4 x p ( t ) + 3 x ( t ) = cos t, x (0) = 0 , x p (0) = 0 . (Note that this is not an equation of a damped vibration as the speed term has a negative coeFcient.) 4. Let A : = 3 3 2 3 3 1 2 2 3 2 4 . (a) (20 points) Determine the eigenvalues of this matrix. (b) (20 points) Show bases for each eigenspace corresponding to the eigenvalues....
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