Quiz4Solutions

Quiz4Solutions - Newberger Math 247 Spring 03 Quiz 1.8 name...

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Unformatted text preview: Newberger Math 247 Spring 03 Quiz 1.8 name: 1. Let T be the transformation given by T ( x ) = A x , where A = 1 2 4 8 0 1 0 0 (a) (4 points) What must a and b be so that T : R a → R b ? The number of entries in x must match the number of columns of A for the definition T ( x ) = A x to make sense. So a = 2 . Similarly, since A x is a linear combination of the columns of A , T ( x ) is a linear combination of the columns of A , and hence for each x , T ( x ) is a vector with the same number of rows as A . So b = 4 . (b) (5 points) Let b = 3 12 3 . Is b in the range of T ? How do you know? We want to know if there is a vector x such that T ( x ) = b , so we solve the equation A x = b , given by the augmented matrix below. 1 2 3 4 8 12 0 1 3 0 0 ∼ 1 2 3 0 1 3 0 0 0 0 Since this system is consistent, there is at least one vector x such that A x = b , and hence by the definition of...
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Quiz4Solutions - Newberger Math 247 Spring 03 Quiz 1.8 name...

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