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Unformatted text preview: (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....
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This note was uploaded on 03/23/2010 for the course MAT 022A taught by Professor Pon during the Spring '10 term at UC Davis.
- Spring '10