Let & denote the vector space of all functions &: R&R under the usual operations of function addition and scalar multiplication of functions. Let It be the subspace of I sponned by the two functions sint and cost, and let B = { sint, costs, which is a basis for H. a) Let T: HH be defined by T(F) = f'. ie, I takes the first derivative. Find the matrix [T]B for the linear transformation T relative to the basis B. b) Let f(t) = 4 cos (t-3). Use the trigonometric identity COS (CA-B) = cos A cos B + Sin A sin B to show that f is in the subspace H. Then give the coordinate Vector []B. C) Let K= Span (sint, cost, 4 cos (t-24)}, which is a subspace of F. Give dim k.
Let & denote the vector space of all functions &: R&R under the usual operations of function addition and scalar multiplication of functions. Let It be the subspace of I sponned by the two functions sint and cost, and let B = { sint, costs, which is a basis for H. a) Let T: HH be defined by T(F) = f'. ie, I takes the first derivative. Find the matrix [T]B for the linear transformation T relative to the basis B. b) Let f(t) = 4 cos (t-3). Use the trigonometric identity COS (CA-B) = cos A cos B + Sin A sin B to show that f is in the subspace H. Then give the coordinate Vector []B. C) Let K= Span (sint, cost, 4 cos (t-24)}, which is a subspace of F. Give dim k.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let denote the vector space of all functions f:R-SR, under
the usual operations of function addition and scalar multiplication
of functions. Let It be the subspace of I spanned by the two
functions sint and cost, and let B = { sint, costs, which is a
basis for H.
a) Let T: HH be defined by T(F) = f'. ie, I takes the first
derivative. Find the matrix [T]B for the linear transformation
f
T relative to the basis B.
b) Let f(t) = 4 cos (t-31). Use the trigonometric identity
COS (A-B) = COS A cos B + Sin A sin B to show that of is in
the subspace H. Then give the coordinate Vector [$]B.
() Let K = Spon (sint, cost, 4 cos (t-24) {, which is a subspace
of F. Give dim K.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee563bfe-badf-4a9d-9a7e-d47066dc4e0e%2F6d380758-7dbc-41a3-893a-4959d2534138%2Feapl73k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let denote the vector space of all functions f:R-SR, under
the usual operations of function addition and scalar multiplication
of functions. Let It be the subspace of I spanned by the two
functions sint and cost, and let B = { sint, costs, which is a
basis for H.
a) Let T: HH be defined by T(F) = f'. ie, I takes the first
derivative. Find the matrix [T]B for the linear transformation
f
T relative to the basis B.
b) Let f(t) = 4 cos (t-31). Use the trigonometric identity
COS (A-B) = COS A cos B + Sin A sin B to show that of is in
the subspace H. Then give the coordinate Vector [$]B.
() Let K = Spon (sint, cost, 4 cos (t-24) {, which is a subspace
of F. Give dim K.
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