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Math 3330 Matices and Linear Algebra Final Solutions

Math 3330 Matices and Linear Algebra Final Solutions - Math...

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Math 3330 SOLUTIONS FOR PRACTICE FINAL EXAM 1. Augmented matrix: A = 1 2 0 - 1 2 2 3 - 1 1 4 0 - 1 - 1 3 0 . Gaussian elimination: 1 2 0 - 1 2 2 3 - 1 1 4 0 - 1 - 1 3 0 R 2 R 2 - 2 R 1 1 2 0 - 1 2 0 - 1 - 1 3 0 0 - 1 - 1 3 0 R 3 R 3 - R 2 1 2 0 - 1 2 0 - 1 - 1 3 0 0 0 0 0 0 . Pivot variables: x and y , non-pivot variables: z and w . Let z = s , w = t . Then y = - s + 3 t , x = 2 + 2 s - 5 t . In paramtertic form, x y w z = 2 0 0 0 + s 2 - 1 1 0 + t - 5 3 0 1 . The spanning vectors are 2 - 1 1 0 , - 5 3 0 1 and the translation vector, 2 0 0 0 . 2. W contains the 0 function. Let f and g be two functions from W . ( f + g )(2) = f (2) + g (2) = 0 + 0 = 0, thus f + g lies in W as well. If f is a function from W and c any constant, then ( cf )(2) = cf (2) = 0, thus cf lies in W as well. 3. The dependence equation is 1 2 - 1 1 x + 2 - 3 - 2 - 5 y + 0 1 2 5
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