Let V be a vector space, and T :V →V a linear transformation such that T(5v1 + 302) = 40, – 402 and T(301 + 202) = 201 + 579. Then V2, T(3,) T(v2) T(201 – 202) =
Let V be a vector space, and T :V →V a linear transformation such that T(5v1 + 302) = 40, – 402 and T(301 + 202) = 201 + 579. Then V2, T(3,) T(v2) T(201 – 202) =
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 \( V \) be a vector space, and \( T : V \rightarrow V \) a linear transformation such that
\[ T(5\vec{v}_1 + 3\vec{v}_2) = 4\vec{v}_1 - 4\vec{v}_2 \]
and
\[ T(3\vec{v}_1 + 2\vec{v}_2) = 2\vec{v}_1 + 5\vec{v}_2. \]
Then
\[
T(\vec{v}_1) = \, \underline{\hspace{3em}} \, \vec{v}_1 + \underline{\hspace{3em}} \, \vec{v}_2
\]
\[
T(\vec{v}_2) = \, \underline{\hspace{3em}} \, \vec{v}_1 + \underline{\hspace{3em}} \, \vec{v}_2
\]
\[
T(2\vec{v}_1 - 2\vec{v}_2) = \, \underline{\hspace{3em}} \, \vec{v}_1 + \underline{\hspace{3em}} \, \vec{v}_2.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d032d57-b5f9-4dc0-88bd-1c45c5e49abc%2Fd46a85be-8426-438c-8d95-d570ea40ac04%2Fze2diwf_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( V \) be a vector space, and \( T : V \rightarrow V \) a linear transformation such that
\[ T(5\vec{v}_1 + 3\vec{v}_2) = 4\vec{v}_1 - 4\vec{v}_2 \]
and
\[ T(3\vec{v}_1 + 2\vec{v}_2) = 2\vec{v}_1 + 5\vec{v}_2. \]
Then
\[
T(\vec{v}_1) = \, \underline{\hspace{3em}} \, \vec{v}_1 + \underline{\hspace{3em}} \, \vec{v}_2
\]
\[
T(\vec{v}_2) = \, \underline{\hspace{3em}} \, \vec{v}_1 + \underline{\hspace{3em}} \, \vec{v}_2
\]
\[
T(2\vec{v}_1 - 2\vec{v}_2) = \, \underline{\hspace{3em}} \, \vec{v}_1 + \underline{\hspace{3em}} \, \vec{v}_2.
\]
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