e1-sample

# e1-sample - A = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4...

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Math 204. Exam 1 (practice) September 17th, 2010 This is the practice exam. The actual exam is about the same length. Note that this is supposed to give you a feeling for what the actual exam will be like. It does not represent every “type” of problem. 1. Suppose that ~v 1 ,~v 2 ,~v 3 are three vectors. Is it possible for any pair of them to be linearly independent, while all three are linearly dependent? If yes, give an example and justify your claims. If no, prove it. 2. Decide whether each of the following sets S is a subspace. If it is, ﬁnd a basis. If not, explain why? (a) Let ~v R n , and S = { ~x R n : ~x · ~v = 1 } (b) The set S of vectors ~x R n , n 3, that satisfy x 1 - 3 x 2 = 0, x 2 + x 3 = 0. 3. Find a basis for the range and kernel of the matrix A (shown below). Also state the dimensions of these two subspaces.
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Unformatted text preview: A = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 . 4. Find a polynomial p of degree 3 that passes through the points (1 , 0), (-1 , 0) and has p (0) = 0. How many such polynomials are there. Sketch one of them. 5. Consider the linear system A~x = ~ b , where A = 1 a 1 2 b 1-2 c . For which choices of a,b,c does the system always have a unique solution. For the other choices of a,b,c give examples to show that the system may have either inﬁnitely many or no solutions. 6. Suppose that ~v 1 ,...,~v n are n vectors in R n . Prove the following theorem: If ~v i · ~v j = 0 for 1 ≤ i < j ≤ n , then { ~v 1 ,...,~v n } is linearly indepedent. 1...
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