gilabert1985
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Hi, I have the following problem that is part of a project, and I have been stuck on it for the last couple of hours ... Thanks a lot for any help you can give me!
It says:
"An affine mapping from V to W, where W is a second real vector space, is a mapping f:V[itex]\rightarrow[/itex]W such that
for all x,y[itex]\epsilon[/itex]V and all [itex]\lambda\epsilon\Re[/itex].
QUESTION:
Prove that f:V[itex]\rightarrow[/itex]W is affine if and only if it is of the form
f(x)=Tx+b, ---> (this is formula (1))
where T:V[itex]\rightarrow[/itex]W is linear and b[itex]\epsilon[/itex]W. [Hint: to prove the 'only if' part, consider the mapping f-f(0).]
Show further that T and b in (1) are uniquely determined by f."
So I know the definition of affine functions from Wikipedia (http://en.wikipedia.org/wiki/Affine_transformation), so the hint I get that suggests me to consider the mapping f-f(0) makes sense. However, I have no clue how to do it without numbers (like it is done in the examples of page 7 and 8 of these notes http://cfsv.synechism.org/c1/sec15.pdf)
It says:
"An affine mapping from V to W, where W is a second real vector space, is a mapping f:V[itex]\rightarrow[/itex]W such that
for all x,y[itex]\epsilon[/itex]V and all [itex]\lambda\epsilon\Re[/itex].
QUESTION:
Prove that f:V[itex]\rightarrow[/itex]W is affine if and only if it is of the form
f(x)=Tx+b, ---> (this is formula (1))
where T:V[itex]\rightarrow[/itex]W is linear and b[itex]\epsilon[/itex]W. [Hint: to prove the 'only if' part, consider the mapping f-f(0).]
Show further that T and b in (1) are uniquely determined by f."
So I know the definition of affine functions from Wikipedia (http://en.wikipedia.org/wiki/Affine_transformation), so the hint I get that suggests me to consider the mapping f-f(0) makes sense. However, I have no clue how to do it without numbers (like it is done in the examples of page 7 and 8 of these notes http://cfsv.synechism.org/c1/sec15.pdf)