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Prove that if T:V-oW io a one to-one linear trans formation and 33 o a linearly deperdent then {TC,), T3,), set in V, then {T,TQ),...,TI} linearly independet set in W, (* Start with the test equation to show that the orly possible solution isc, =C2=.... =ek=0.) こ

Prove that if T:V-oW io a one to-one linear trans formation and 33 o a linearly deperdent then {TC,), T3,), set in V, then {T,TQ),...,TI} linearly independet set in W, (* Start with the test equation to show that the orly possible solution isc, =C2=.... =ek=0.) こ

Advanced Engineering Mathematics
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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Prove that if \( T: V \rightarrow W \) is a one-to-one linear transformation and \(\{\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_k\}\) is a linearly dependent set in \( V \), then \(\{T(\vec{v}_1), T(\vec{v}_2), \ldots, T(\vec{v}_k)\}\) is a linearly independent set in \( W \). (* Start with the test equation \( c_1 T(\vec{v}_1) + c_2 T(\vec{v}_2) + \ldots + c_k T(\vec{v}_k) = \vec{0}_W \) to show that the only possible solution is \( c_1 = c_2 = \ldots = c_k = 0 \).)
Transcribed Image Text:Prove that if \( T: V \rightarrow W \) is a one-to-one linear transformation and \(\{\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_k\}\) is a linearly dependent set in \( V \), then \(\{T(\vec{v}_1), T(\vec{v}_2), \ldots, T(\vec{v}_k)\}\) is a linearly independent set in \( W \). (* Start with the test equation \( c_1 T(\vec{v}_1) + c_2 T(\vec{v}_2) + \ldots + c_k T(\vec{v}_k) = \vec{0}_W \) to show that the only possible solution is \( c_1 = c_2 = \ldots = c_k = 0 \).)
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