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# 04spex2 - show a nontrivial linear combination resulting in...

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Math 320 Second In-class Exam April 13, 2004 M´arton Bal´azs NAME: 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 2 × 2 matrices for which A 2 = 0 . (b) (15 points) S 2 is the set of B 2 × 2 matrices for which B T = B .

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2. (40 points) By computing the Wronskian, decide if ln( x ) , ln(2 x ) , ln(3 x ) are linearly inde- pendent functions on the real interval (0 , ). Bonus question (5 points): If they are independent, then show directly that all linear combi- nations resulting in the zero function must have zero coefficients. If they are dependent, then

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Unformatted text preview: show a nontrivial linear combination resulting in the zero function. 3. (50 points) Solve the initial value problem x pp ( t ) + 2 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 spring force term has a negative coeFcient.) 4. Let A : = 1 1 1 1 1 1 1 1 1 . (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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