2. (a) Use the linearity of matrix-vector multiplication to prove the following: if the vectors x1, X2 are solutions to Ax = 0 (3.1) then the linear combination x = c¿x1 + c2X2 is also a solution, for any constants c1, c2. (You may use the results you showed in the previous problem) (b) Show that the results is not true for nonhomogeneous systems: if the vectors x1, x2 are solutions to Ax = b (3.2) with b + 0, then the linear combination x = c]x1 + C2X2 is not a solution. (c) Show that if, however, x1 and x2 are solutions to the homogenous problem (3.1) and x, is a particular solution to the nonhomogeneous problem (3.2), then

2. (a) Use the linearity of matrix-vector multiplication to prove the following: if the vectors x1, X2 are solutions to Ax = 0 (3.1) then the linear combination x = c¿x1 + c2X2 is also a solution, for any constants c1, c2. (You may use the results you showed in the previous problem) (b) Show that the results is not true for nonhomogeneous systems: if the vectors x1, x2 are solutions to Ax = b (3.2) with b + 0, then the linear combination x = c]x1 + C2X2 is not a solution. (c) Show that if, however, x1 and x2 are solutions to the homogenous problem (3.1) and x, is a particular solution to the nonhomogeneous problem (3.2), then

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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Question

2. (a) Use the linearity of matrix-vector multiplication to prove the following: if the vectors x1,
X2 are solutions to
Ах
= 0
(3.1)
then the linear combination x = cịx1 + c2x2 is also a solution, for any constants c1, c2.
(You may use the results you showed in the previous problem)
(b) Show that the results is not true for nonhomogeneous systems: if the vectors x1, X2 are
solutions to
Ax = b
(3.2)
with b + 0, then the linear combination x = c1x1 + c2x2 is not a solution.
(c) Show that if, however, x1 and x2 are solutions to the homogenous problem (3.1) and
is a particular solution to the nonhomogeneous problem (3.2), then
Хр
х— С1X1 + C2X2 + Хр
is a solution for the nonhomogeneous problem (3.2), for any c1, C2.

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