- #1
TheRookie
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ℝ
I'm a 2nd year undergraduate student, so I suppose many users here won't find this too difficult, but I've had some issues with the following questions and, of course, any help would be very much appreciated:
(i) Prove f: V→W is affine (where V and W are real vector space) iff it is of the form f(x) = Tx + b, where T: V→ W is linear and b∈W. Prove T and b are uniquely determined by f.
(ii) Prove the image of an affine subset under an affine mapping is affine. Prove the composition of two affine maps is affine.
(i) If f: V→W is an affine mapping, we must have f(λx + (1-λ)y) = λf(x) + (1-λ)f(y) for all x and y in V, and λ in ℝ.
This is what immediately came to mind, but I can't see a connection between this and the question...
(ii) I'm sorry, but I've gone over this again and again and don't know where to start.Regards,
Pete
Homework Statement
I'm a 2nd year undergraduate student, so I suppose many users here won't find this too difficult, but I've had some issues with the following questions and, of course, any help would be very much appreciated:
(i) Prove f: V→W is affine (where V and W are real vector space) iff it is of the form f(x) = Tx + b, where T: V→ W is linear and b∈W. Prove T and b are uniquely determined by f.
(ii) Prove the image of an affine subset under an affine mapping is affine. Prove the composition of two affine maps is affine.
Homework Equations
The Attempt at a Solution
(i) If f: V→W is an affine mapping, we must have f(λx + (1-λ)y) = λf(x) + (1-λ)f(y) for all x and y in V, and λ in ℝ.
This is what immediately came to mind, but I can't see a connection between this and the question...
(ii) I'm sorry, but I've gone over this again and again and don't know where to start.Regards,
Pete