# 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⊕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

....

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

#### 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.