Linear Transformations: the following ask you to work with the two defining properties of linear transformations. (a) Show that D, defined as is is a linear transformation. (b) Show that I, defined as d D(y(t)) = [y(t)] + p(t)y(t), I(g(t)) - 19 = g(t) dt, is a linear transformation. If the input is g(t), describe the kinds of functions that are outputs of I(g(t)). (c) Show that L, defined as L (f (t)) = e-st -st f(t) dt, is a linear transformation. If the input is f(t), describe the kinds of functions that are outputs of L(ƒ(t)) – specifically, identify the independent variable for the output function.
Linear Transformations: the following ask you to work with the two defining properties of linear transformations. (a) Show that D, defined as is is a linear transformation. (b) Show that I, defined as d D(y(t)) = [y(t)] + p(t)y(t), I(g(t)) - 19 = g(t) dt, is a linear transformation. If the input is g(t), describe the kinds of functions that are outputs of I(g(t)). (c) Show that L, defined as L (f (t)) = e-st -st f(t) dt, is a linear transformation. If the input is f(t), describe the kinds of functions that are outputs of L(ƒ(t)) – specifically, identify the independent variable for the output function.
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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![### Linear Transformations
The following exercises ask you to work with the two defining properties of linear transformations.
**(a)** Show that \( D \), defined as
\[
D(y(t)) = \frac{d}{dt} \left[ y(t) \right] + p(t)y(t),
\]
is a linear transformation.
**(b)** Show that \( I \), defined as
\[
I(g(t)) = \int_a^b g(t) \, dt,
\]
is a linear transformation. If the input is \( g(t) \), describe the kinds of functions that are outputs of \( I(g(t)) \).
**(c)** Show that \( \mathcal{L} \), defined as
\[
\mathcal{L}(f(t)) = \int_0^\infty e^{-st} f(t) \, dt,
\]
is a linear transformation. If the input is \( f(t) \), describe the kinds of functions that are outputs of \( \mathcal{L}(f(t)) \) – specifically, identify the independent variable for the output function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa40558bc-afda-4e2b-be1a-c4211c27f73c%2F7fdcb295-d900-422e-82dc-70a860adc6b0%2Fcuoiy0s_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformations
The following exercises ask you to work with the two defining properties of linear transformations.
**(a)** Show that \( D \), defined as
\[
D(y(t)) = \frac{d}{dt} \left[ y(t) \right] + p(t)y(t),
\]
is a linear transformation.
**(b)** Show that \( I \), defined as
\[
I(g(t)) = \int_a^b g(t) \, dt,
\]
is a linear transformation. If the input is \( g(t) \), describe the kinds of functions that are outputs of \( I(g(t)) \).
**(c)** Show that \( \mathcal{L} \), defined as
\[
\mathcal{L}(f(t)) = \int_0^\infty e^{-st} f(t) \, dt,
\]
is a linear transformation. If the input is \( f(t) \), describe the kinds of functions that are outputs of \( \mathcal{L}(f(t)) \) – specifically, identify the independent variable for the output function.
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