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# Vector Spaces and Projection Mappings

Please see the attached file for the fully formatted problems.

Let V be a vector space of all real continuous function on closed interval [ -1, 1]. Let Wo be a set of all odd functions in V and let We be a set of all even functions in V.
(i) Show that Wo and We are subspaces and then show that V=Wo&#8853;We.
(ii) Find a projection mapping onto Wo parallel to We and projection mapping onto We parallel to Wo.
(iii) Let L: V -> V be a mapping that transforms f from V into function that is given by
....
L f x = &#8747; f t dt
Show that L is linear mapping and state whether the following is true or false:
[ ]LWo&#8834;We and [ ]LWe&#8834;Wo
Solution:
(i) V=Wo&#8853;We then every element in V can be written as
...
even function odd function
fx fx f x fx f x
function even function odd function
.....
(ii) To find a projection mapping we use:
N and P are R -submodules of M such that M =N&#8853;P and &#981; that is
projection mapping onto N parallel to P. Then
....

#### Solution Summary

Vector spaces and projection mappings are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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