e406k - with your approximate semester grade so far. The...

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MAS 3105 June 1, 2006 Quiz 4 and Key Prof. S. Hudson 1) Use the Wronskian to decide whether the functions { x + 2 ,x 2 - 1 } are LI in P 3 . Then comment briefly on your reasoning. 2) Let S R 3 be the set of vectors of the form [ a + 2 b + c, 2 a + 4 b,a + 2 b ] T . Find a basis for S , check whether it really is a basis, and comment. 3) Answer True or False: The columns of a nonsingular matrix are always linearly independent. If L is a list of n vectors, then dim (span L ) = n . If A is a singular 4x4 matrix, then its nullity is at least 1. R 2 is a subspace of R 3 . If L is a list of 3 L.I. functions in P 3 , then span ( L ) = P 3 . 4) Choose ONE of these to prove (on the back is OK). a) If U and V are subspaces of a vector space W , then U V is too. b) [This is part of Thm.3.3.2] Suppose L is a linearly dependent list of vectors, and v span ( L ). Then v is a linear combination of the vectors in L in at least two different ways. Remarks and Answers: The problems were worth 15 points each. The average was about 40/60 (similar to Quiz 2). I wrote your Quiz 1-4 total “Σ” in the upper right corner,
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Unformatted text preview: with your approximate semester grade so far. The A-’s started at about 200, and each letter is 25 points. Later I will include your HW, and will replace your lowest quiz score with your average MHW grade. 1) W = det ± x + 2 x 2-1 1 2 x ² = x 2 + 4 x + 1 6 = 0 . So , they are LI . 2) Your final answer should be a basis of S ; a list of LI vectors in R 3 . Not a matrix, not a scalar, etc. This problem is very similar to HW 3.4.7 (factor out the a, b, c), but since the 3 vectors you get that way are LD, you must remove one (-3 points if you didn’t). a + 2 b + c 2 a + 4 b a + 2 b = a 1 2 1 + b 2 4 2 + c 1 So, one basis for S is { [1 2 1] T , [1 0 0] T } . 3) TFTFT 4) See lectures and the text. 1...
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This note was uploaded on 12/26/2011 for the course MAS 3105 taught by Professor Julianedwards during the Spring '09 term at FIU.

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