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    Vector Spaces and Projection Mappings

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    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⊕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 = ∫ f t dt
    Show that L is linear mapping and state whether the following is true or false:
    [ ]LWo⊂We and [ ]LWe⊂Wo
    Solution:
    (i) V=Wo⊕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⊕P and ϕ that is
    projection mapping onto N parallel to P. Then
    ....

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    https://brainmass.com/math/vector-calculus/vector-spaces-projection-mappings-118459

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