Let 33,3 be a basio for a rector space v and suppose that T,:VV and BiV V are linear transformations satisfying Tz Q,)=-5, Furid the rule for (TzT,X)for every vector DEV always be (* Use the faet that J can written in terms of the basis rectors
Let 33,3 be a basio for a rector space v and suppose that T,:VV and BiV V are linear transformations satisfying Tz Q,)=-5, Furid the rule for (TzT,X)for every vector DEV always be (* Use the faet that J can written in terms of the basis rectors
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 \(\{ \vec{v}_1, \vec{v}_2 \}\) be a basis for a vector space \( V \) and suppose that \( T_1: V \to V \) and \( T_2: V \to V \) are linear transformations satisfying:
\[
T_1(\vec{v}_1) = \vec{v}_1 - \vec{v}_2
\]
\[
T_1(\vec{v}_2) = 3\vec{v}_1 + \vec{v}_2
\]
\[
T_2(\vec{v}_1) = -5\vec{v}_2
\]
\[
T_2(\vec{v}_2) = -\vec{v}_1 + 2\vec{v}_2
\]
Find the rule for \((T_2 \circ T_1)(\vec{v})\) for every vector \(\vec{v} \in V\).
(* Use the fact that \(\vec{v}\) can always be written in terms of the basis vectors for \( V \).)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe613d3f6-ed3c-4abe-81e0-270d8219b401%2F4ce00d20-7000-4e90-8f17-ee5349bca78f%2F7wpsqbz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let \(\{ \vec{v}_1, \vec{v}_2 \}\) be a basis for a vector space \( V \) and suppose that \( T_1: V \to V \) and \( T_2: V \to V \) are linear transformations satisfying:
\[
T_1(\vec{v}_1) = \vec{v}_1 - \vec{v}_2
\]
\[
T_1(\vec{v}_2) = 3\vec{v}_1 + \vec{v}_2
\]
\[
T_2(\vec{v}_1) = -5\vec{v}_2
\]
\[
T_2(\vec{v}_2) = -\vec{v}_1 + 2\vec{v}_2
\]
Find the rule for \((T_2 \circ T_1)(\vec{v})\) for every vector \(\vec{v} \in V\).
(* Use the fact that \(\vec{v}\) can always be written in terms of the basis vectors for \( V \).)
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