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Applied Linear Algebra

by: Mary Veum

Applied Linear Algebra MATH 2210

Mary Veum
GPA 3.97

Gerald Leibowitz

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Gerald Leibowitz
Class Notes
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This 1 page Class Notes was uploaded by Mary Veum on Thursday September 17, 2015. The Class Notes belongs to MATH 2210 at University of Connecticut taught by Gerald Leibowitz in Fall. Since its upload, it has received 31 views. For similar materials see /class/205827/math-2210-university-of-connecticut in Mathematics (M) at University of Connecticut.


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Date Created: 09/17/15
Mathematics 2210 Example Is T oneto one or onto 1 0 Consider the function T with the rule T 2 1 3 2 ls T one to one ls the range of T all of R3 That is7 is T onto lf not7 what is the range of T We can answer these questions directly or we can try to apply the theory from Section 19 It is clear that there are vectors other than the zero vector at which T 0 has the value 0 eg7 T 0 0 no matter what 3 is So T is not 3 one to one Likewise7 it is clear that all of the values of T have rst coordinate equal to zero7 so not every vector in R3 belongs to the range of T Thus T isn7t onto To nd the exact range of a function is often a dif cult task7 but in this example it isnt hard lf b E R3 is in the range of T7 then b1 0 But conversely7 if b1 07 then we can solve the equation TX b7 which is the system of equations 0 b1 1 b2 2 b3 and see that there are solutions The general solution is x bg7 baht with t arbitrary7 ie7 any real number So every such b is a member of the range Conclusion In this example7 the range of the transformation is the set of all b E R3 for which b1 0 ie7 RangeT Span627 63 Since the matrix of T is 000 A100 010 Explain why this is so7 we could note that the columns of A are linearly dependent since one of them is a zero vector and not every row ofA contains a pivot position Hence by theorems in Sec 197 T isn7t one to one or onto


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