Dual of Linear Map

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Generic Goon
Posts: 18
Joined: Sun Jan 27, 2008 5:20 pm UTC

Dual of Linear Map

Postby Generic Goon » Thu Sep 30, 2010 7:09 pm UTC

(For homework, but just need to understand the question.)

I want to show if T: V -> W is linear, then T*: W* -> V* is linear. To show T* has additive property, do I need to show that:

T*(g+g')(v) = T*g(v) + T*g'(v) (Where g and g' are functionals), or do I need to show:

T*g(v+v') = T*g(v) + T*g(v') (Where v and v' are vectors).

I'm pretty sure that the methods for showing each are almost identical, but I don't know which to actually do.

mark999
Posts: 75
Joined: Fri Jul 11, 2008 1:00 am UTC

Re: Dual of Linear Map

Postby mark999 » Thu Sep 30, 2010 7:12 pm UTC

It's been a while since I've done this stuff, but I'm almost certain it's the first one. The second one is saying that T*g is additive.

mike-l
Posts: 2758
Joined: Tue Sep 04, 2007 2:16 am UTC

Re: Dual of Linear Map

Postby mike-l » Thu Sep 30, 2010 7:38 pm UTC

Yes, definitely the first one. The 'vectors' in W* are the functionals.
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