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HW3sol - (Partial Solutions to Homework 3 Problem 7.1 Show...

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(Partial) Solutions to Homework 3 Problem 7.1: Show that the pair of conditions: L ( u + v ) = L ( u ) + L ( v ) L ( cv ) = cL ( v ) is equivalent to the single condition: L ( ru + sv ) = rL ( u ) + sL ( v ) Your answer should have two parts. Show that (1 , 2) = (3) and then that (3) = (1 , 2). Answer: In general I’ve talked a lot about this problem already, so I won’t go into too much explanation. But it might be helpful to see another example of how to write a proof. This is something like what I would want to see: Assume first that L ( u + v ) = L ( u ) + L ( v ) for all u, v V and that L ( cv ) = cL ( v ) for all c R , v V . Then we have that L ( ru + sv ) = L ( ru ) + L ( sv ) by condition (1), and that L ( ru ) = rL ( u ) and L ( sv ) = sL ( v ) by condition (2). Putting these together, we have L ( ru + sv ) = L ( ru ) + L ( sv ) = rL ( u ) + sL ( v ), so (3) holds. Next, assume that L ( ru + sv ) = rL ( u ) + sL ( v ) for all r, s R and u, v V . Then in particular, (3) holds when r = s = 1. In this case, we have L ( u + v ) = L ( u )+ L ( v ), so (1) holds. Similarly, (3) must hold for if s = 0, which gives us L ( ru ) = L ( ru + 0 v ) = rL ( u ) + 0 L ( v ) = rL ( u ), so (2) holds. Therefore, (1,2) is equivalent to (3).
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