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dkotschessaa
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I'm confused about some of the notation in Hoffman & Kunze Linear Algebra.
Let V be the set of all complex valued functions f on the real line such that (for all t in R)
[itex] f(-t) = \overline{f(t)}[/itex] where the bar denotes complex conjugation.
Show that V with the operations (f+g)(t) = f(t) + g(t)
(cf)(t) = cf(t) is a vector space over the field of real numbers. Give an example of a function in V which is not real valued.
Before I can even approach the question I need to clarify what is happening here.
I'm not sure why it is written as ##f(-t) = \overline{f(t)} ##
Am I correct that ## f(a) = a - 0i = a ## (since a is real).
## f(-a) ## would be ## a + 0i = a ##
Obviously I have some confusion here.
Also, with the addition properties given above, would I not have:
## (f+g)(t) = f(t) + g(t) = \overline{f(-t)} + \overline{g(-t)} ##
In particular I am confused when I try to show closure under the real numbers. since ##(a_{1} + b_{1}i) + (a_{2} + b_{2}i)## gives me (I think) something like ## a_{1} + a_{2} + (b_{1} + b_{2})i ##Unless I'm supposed to be adding something to it's OWN conjugation, in which case the imaginary part would cancel, which would be nice. Is that what I am supposed to do?
Appreciate any help
-Dave K
Let V be the set of all complex valued functions f on the real line such that (for all t in R)
[itex] f(-t) = \overline{f(t)}[/itex] where the bar denotes complex conjugation.
Show that V with the operations (f+g)(t) = f(t) + g(t)
(cf)(t) = cf(t) is a vector space over the field of real numbers. Give an example of a function in V which is not real valued.
Before I can even approach the question I need to clarify what is happening here.
I'm not sure why it is written as ##f(-t) = \overline{f(t)} ##
Am I correct that ## f(a) = a - 0i = a ## (since a is real).
## f(-a) ## would be ## a + 0i = a ##
Obviously I have some confusion here.
Also, with the addition properties given above, would I not have:
## (f+g)(t) = f(t) + g(t) = \overline{f(-t)} + \overline{g(-t)} ##
In particular I am confused when I try to show closure under the real numbers. since ##(a_{1} + b_{1}i) + (a_{2} + b_{2}i)## gives me (I think) something like ## a_{1} + a_{2} + (b_{1} + b_{2})i ##Unless I'm supposed to be adding something to it's OWN conjugation, in which case the imaginary part would cancel, which would be nice. Is that what I am supposed to do?
Appreciate any help
-Dave K
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