Prove that the given transformation is a linear transformation, using the definition (or the Remark following Example 3.55). x + y х — у Tis a linear transformation if and only if X2 T(C,V, + c,v2) = c,T(V;) + c,T(V2), where v, V2 Then, entering your answers in terms of c,, C2, x1, X2• Y1v and y2 X2 C2 X1 T(C,V1 + Cv2) Y1 = T C2X2 + C2Y2 C;X1 - C1Y1 X2 + Y2 X1- Y1 = c,T(v,) + c2T(v2) %3! and thus T is linear.
Prove that the given transformation is a linear transformation, using the definition (or the Remark following Example 3.55). x + y х — у Tis a linear transformation if and only if X2 T(C,V, + c,v2) = c,T(V;) + c,T(V2), where v, V2 Then, entering your answers in terms of c,, C2, x1, X2• Y1v and y2 X2 C2 X1 T(C,V1 + Cv2) Y1 = T C2X2 + C2Y2 C;X1 - C1Y1 X2 + Y2 X1- Y1 = c,T(v,) + c2T(v2) %3! and thus T is linear.
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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Transcribed Image Text:Prove that the given transformation is a linear transformation, using the definition (or the Remark following Example 3.55).
Tis a linear transformation if and only if
T(c,v, + czv2) = c,T(v,) + c,T(v,), where v, =
v, =
Then, entering your answers in terms of c,, C2, X1, X2, Y,, and y,
T(c,v, + c2v2) =
+ Ca
Y2
C,
=
C2X2 + C2Y2
CX1 - C1Y1
X2 + Y2
= C1
+ C2
X1 - Y1
= cT(v,) + c2T(v2)
and thus Tis linear.
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