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MAT485-2007Springa

# MAT485-2007Springa - M AT 485 Final Exam 9 May 2007 Prof V...

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MAT 485 Final Exam 9 May 2007 Prof. V. Fatica Name _ SUID _

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1. Solve the initial value problem: 4t 3 y' = (y _1)2' y(O) = 3
= 5 2. Consider the system of equations: 3x 1 +X 2 -X 3 +2x 3 -2x 4 -5x 4 = = o 15 2x] 2 + 3x 3 =10 (a) Write the system as an augmented matrix. (b) Reduce the augmented matrix to reduced row echelon form. Xl X 2 (c) Give the solutions to the system in parametric vector form: = X 3 X 4 (d) From the six alternatives below, choose (circle) the one that describes the solution set (a subset of R 4 ) ofthe system of equations: EMPTY A SINGLE POINT ALINE A PLANE A 3-D HYPERPLANE ALL OF R 4

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3. Consider the linear transformation T:RJ_R J defined by [[ X]] [X + 2y + 3Z] T(X)=T y = x+3y+4z . Z X+ y+2z (a) Find a matrix A such that T(X) = AX . (b) Determine the dimension of Ker(1). (c) Give an independent set of vectors that spans Ker(1). (d) What is Dim(Im(1))? (e) Give an independent set ofvectors that spans Im(1). (f) Is T injective (one-to-one)? Circle one: YES (g) Is T surjective (onto)? Circle one: YES NO NO
4. Consider the critically damped, forced harmonic oscillator governed by the differential

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MAT485-2007Springa - M AT 485 Final Exam 9 May 2007 Prof V...

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