Quiz2-Solutions

# Quiz2-Solutions - R 4 Solution B Another way to solve the...

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Name:. ......................................................................................... Math 331 - Quiz 2 Thursday, September 15 1. Construct a 3 × 3 matrix A with nonzero entries and a vector b in R 3 such that b is not in the set spanned by the columns of A . Justify your answer. Solution: There are many possible answers to this question. One is A = 1 1 1 1 1 1 1 1 1 b = 1 2 3 The columns of A are all the same vector , 1 1 1 . So the span of the columns are just all vectors of the form c 1 1 1 = c c c . In other words, the span consists of vectors with all three entries the same. so 1 2 3 is not in the span. 2. Let v 1 = 1 0 - 1 0 v 2 = 0 - 1 0 1 v 3 = 1 0 0 - 1 Does { v 1 , v 2 , v 3 } span R 4 ? Why or why not? Solution A: It does not span R 4 . The easiest way to see this is Theorem 4 from the book which says that { v 1 , v 2 , v 3 } spans R 4 if and only if every row in the matrix formed by the columns of A contains a pivot position. Since there are three vectors, there are at most three pivot columns. However there are four rows, so there must be at least one row with out a pivot. Therefore the vectors can not span

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Unformatted text preview: R 4 . Solution B: Another way to solve the problem is to do it directly. Let b = b 1 b 2 b 3 b 4 be a vector in R 4 . b is in the span of { v 1 , v 2 , v 3 } if the system 1 1 b 1-1 b 2-1 b 3 1-1 b 4 has a solution. We can row reduce the matrix. 1 1 b 1-1 b 2-1 b 3 1-1 b 4 III + I---→ 1 1 b 1-1 b 2 1 b 1 + b 3 1-1 b 4 IV + II----→ 1 1 b 1-1 b 2 1 b 1 + b 3-1 b 2 + b 4 IV + III----→ 1 1 b 1-1 0 b 2 1 b 1 + b 3 b 1 + b 2 + b 3 + b 4 Therefore the span is the set of vectors where b 1 + b 2 + b 3 + b 4 = 0 Since this is not all of R 4 , the vectors do not span....
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Quiz2-Solutions - R 4 Solution B Another way to solve the...

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