Let V = CR ([-1,1]) denote the vector space of continuous real-value the interval [-1, 1] with inner product defined as (f,g) = [ f(x)g(x)dx. subspace V₁ = Span{ 2, sin(Ta), cos(xx),...,‚ sin(nær), cos(nxx)}. D = L(V₂) by Df = f'. Show that D* = -D. Conclude that D but not self-adioint

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Let V
CR ([-1,1]) denote the vector space of continuous real-valued
functions on the interval [−1, 1] with inner product defined as
(1.9) = [', f(x)g(x)da.
=
Consider the subspace
=
1
, sin(x), cos(x), ..., sin(nx), cos(nTx)}.
√2¹
Vn Span{.
a.) Define D € L(V₂) by Df
normal but not self-adjoint.
b.) Define TE L(V₂) by Tf = f". Show that T is self-adjoint.
=
f'. Show that D*
=
-D. Conclude that D is
Transcribed Image Text:Let V CR ([-1,1]) denote the vector space of continuous real-valued functions on the interval [−1, 1] with inner product defined as (1.9) = [', f(x)g(x)da. = Consider the subspace = 1 , sin(x), cos(x), ..., sin(nx), cos(nTx)}. √2¹ Vn Span{. a.) Define D € L(V₂) by Df normal but not self-adjoint. b.) Define TE L(V₂) by Tf = f". Show that T is self-adjoint. = f'. Show that D* = -D. Conclude that D is
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