Linear algebra prove briefly 3. Let V denote the vector space C[-1, 1] of continuous real-valued functions on the closed interval [-1, 1].So, for example, the functions e and cos(x) – x² belong to this vector space (where both functionsare restricted to the domain [-1,1]). We turn V into an inner product space by defining1(f (x), g(x)) = | f(x)g(x) dx.(a) Let W denote the subspace of odd functions in V (functions that satisfy f(-x) = -f(x) forall x E [-1,1]), and let U denote the subspace of even functions in V (functions that satisfyf(-x) = f(x) for all x E [-1, 1]). Prove that V = W U.W-, with respect to the inner(b) With the help of the result from part (a), prove that in fact Uproduct given in the question.

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Answered: Linear algebra prove briefly

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Linear algebra prove briefly

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 34EQ

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Question

Linear algebra prove briefly

3. Let V denote the vector space C[-1, 1] of continuous real-valued functions on the closed interval [-1, 1].
So, for example, the functions e and cos(x) – x² belong to this vector space (where both functions
are restricted to the domain [-1,1]). We turn V into an inner product space by defining
1
(f (x), g(x)) = | f(x)g(x) dx.
(a) Let W denote the subspace of odd functions in V (functions that satisfy f(-x) = -f(x) for
all x E [-1,1]), and let U denote the subspace of even functions in V (functions that satisfy
f(-x) = f(x) for all x E [-1, 1]). Prove that V = W U.
W-, with respect to the inner
(b) With the help of the result from part (a), prove that in fact U
product given in the question.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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