Vector Spaces and Projection Mappings
Not what you're looking for?
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
....
Purchase this Solution
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.
Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Probability Quiz
Some questions on probability