ch1 - Solutions to Homework 2 Math 110 Fall 2006 Prob...

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Solutions to Homework 2. Math 110, Fall 2006. Prob 1.4.4. (a) Yes, since the linear system a + b = 1 2 a + 3 b = 0 - a = - 3 a - b = 5 has a solution a = 3, b = - 2. (b) No, the corresponding linear system has no solution. (c) Yes, the corresponding linear system has a solution a = 4, b = - 3. (d) Yes, the corresponding linear system has a solution a = - 2, b = 5. (e) No, the corresponding linear system has no solution. (f) No, the corresponding linear system has no solution. Prob 1.4.15. Since any set H is contained in its linear span, we have S 1 S 2 S 1 span( S 1 ), S 1 S 2 S 2 span( S 2 ), hence S 1 S 2 span( S 1 ) span( S 2 ) . (1) The right-hand side of (1) is a subspace, hence taking the span of the left-hand side, we still remain within that subspace, i.e., span( S 1 S 2 ) span( S 1 ) span( S 2 ) . An example of equality is provided by identical sets S 1 = S 2 ; an example of inequality by two vectors which are multiples of each other, say S 1 = { (1 , 0) } , S 2 = { (2 , 0) } . Then S 1 S 2 = , hence span( S 1 S 2 ) = { (0 , 0) } , while span( S 1 ) span( S 2 ) = span( S 1 ) = { ( t, 0) : t IR } . Prob 1.5.1. (a) False: consider a dependent set { v, 0 } where v 6 = 0; v is a not a multiple of 0. (b) True: directly from the definition. (c) False, also directly from the definition. (d) False, take the same example as in (a). (e) True (shown in class). (f) True, from the definition.
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ch1 - Solutions to Homework 2 Math 110 Fall 2006 Prob...

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