prelim 1 - Prelim 1 Math 294 no calculators answer reason =...

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Prelim 1 Math 294 Fall 2007 no calculators answer + reason = credit 1. (10 points) In the study of heat conduction along a metal rod, it is found that a steady state temperature distribution will have the property that for any three consecutive temperatures along the rod, the center one is the average of the other two. For example in the figure, T 1 = 1 2 (20 + T 2 ). (All the labels in the figure are temperatures, including the numbers 20 and 47.) Write another equation relating the various temperatures, then write an augmented matrix for your system and solve it to find the temperatures T 1 and T 2 . T 1 T 2 20 47 2. A linear system Ax = b has augmented matrix which has been reduced to 1 0 2 0 0 : 0 0 1 0 0 4 : 2 0 0 0 1 0 : 6 0 0 0 0 0 : 0 a) (7 points) Find all solution vectors x for the original system. b) (3 points) Do whichever of these can be carried out explicitly: either find a basis for the kernel of A or find a basis for the image of A .
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3. (15 points) A six dimensional subspace V of R 10 has a basis B = { v 1 , v 2 , . . . , v r } . Write v 1 = a 1 a 2 . . . a s and set x = v 3 - 3 v 1 . a) Find r . b) Find s . c) Is 2 . 94 x in V ? d) Is it possible that v 2 = x ? e) Find the coordinate vector [ x ] B . 4. (5 points) Determine whether the vectors w 1 = - 3 1 2 1 , w 2 = - 1 1 1 3 , and w 3 = - 7 - 1 3 - 11 are linearly independent.
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5. (5 points) Let T : R 2 R 2
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