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by: Ms. Helen Sipes


Ms. Helen Sipes
GPA 3.98


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Class Notes
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This 1 page Class Notes was uploaded by Ms. Helen Sipes on Sunday September 27, 2015. The Class Notes belongs to MATH 317 at Iowa State University taught by Staff in Fall. Since its upload, it has received 28 views. For similar materials see /class/214497/math-317-iowa-state-university in Mathematics (M) at Iowa State University.




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Date Created: 09/27/15
Hints for Proofs 1 First write out the hypothesis and conclusion in mathematical terms label variables write out equations etc EXAMPLE Consider the following Theorem The intersection of two subspaces of a vector space is also a subspace Start out by labeling the big vector space by V and the two subspaces by y and Z Hypothesis 3 and Z are subspaces of a vector space V Conclusion 3 Z is also a subspace Of V 2 Assume only the hypothesis and nothing else except previous theorems Don t mix up different types of objects like sets and elements 0quot Proceed logically at each step make sure each step follows from the previous step EXAMPLE The implication y2 32 i y a is not correct for all 331 6 R 5 Use previous theorems and results QTY It may be necessary to divide the proof into several cases as by 0 EXAMPLE Consider the following Theorem lf ad 7 be 31 0 then the system ca dy 0 has exactly one solution a y 0 Proof sketch First suppose a 0 Then b and c are nonzero which implies that y 0 and then a 0 Now suppose that a 31 0 Then a 743 from the rst equation and the proof can be completed by substituting this value into the second equation 6 Maybe use proof by contradiction If you are trying to prove P i Q there are two ways to do this a prove the contrapositive not Q i not P or b assume that P i Q is false ie that P is true and Q is false and proceed until you contradict something such as a previous theorem or some obvious fact like 0 31 1 The following example illustrates a EXAMPLE Consider the following Theorem lf dimV n then any set of n vectors which spans V must also be linearly independent Proof Let S 81 sn be a set of n vectors which spans V and suppose that S is linearly dependent Then by an earlier result we can remove some vector 31 from S such that the resulting set 51 still spans V But the de nition of dimension then implies that dimV S n 7 1 lt n a contradiction to the hypothesis Therefore 8 must be linearly independent Here is a simple example which illustrates EXAMPLE Consider the following Theorem If f is the function de ned by 6 for all a E R then 31 0 for all a E R Proof Suppose fco 0 for some 330 E R Then 0 fa0f7co emoe mo emo mo 60 1 a contradiction Therefore 31 0 for all a E R 7 Look at simple cases to get an idea of why the implication might be true or false Then look at the general case EXAMPLE To show that dimMm mn look at M23 rst 00 Be creative Look at the problem in different ways Maybe write down equivalent statements as b c EXAMPLE The statement the system d y f has a unique solution is equivalent to 3 6y the lines as by c and d3 6y f intersect in a single point 59 Don t give up too easily


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