Explain the determaine 3.2.2Example BConsider the three functionsf1(k) = 3*, f2(k) = k3*, f3(k) = k²3*.(3.42)The linear combinations of these three functions can be written as follows:f1(k) + c2f2(k)+ c3f3(k) = 3* (c1 + c2k + c3k²).(3.43)Note that the right-hand side of equation (3.43) will be zero if and only ifCoC1 + c2k + c3k² = 0.(3.44)But thisan only occur for c1 = C2 = c3 = 0. Therefore, the three functionsgiven in equation (3.42) are linearly independent.An easy calculation shows that the Casoratian for these functions isC(k) = 2 - 33k+3.(3.45)Since C(k) 0, we again conclude that these three functions are linearlyindependent.

Name: Explain the determaine 3.2.2Example BConsider the three functionsf1(k)
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Description: Explain the determaine 3.2.2Example BConsider the three functionsf1(k) = 3*, f2(k) = k3*, f3(k) = k²3*.(3.42)The linear combinations of these three functions can be written as follows:f1(k) + c2f2(k)+ c3f3(k) = 3* (c1 + c2k + c3k²).(3.43)Note that th

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3.2.2 Example B Consider the three functions fi (k) = 3*, f2(k) = k3*, f3 (k) = k23k. (3.42) The linear combinations of these three functions can be written as follows: cifi(k)+ c2f2(k) + c3f3(k) = 3*(c1 + c2k + c3k?). (3.43) Note that the right-hand side of equation (3.43) will be zero if and only if C1 + c2k + c3k2 = 0. (3.44) %3D But this can only occur for cı = c2 = c3 = 0. Therefore, the three functions given in equation (3.42) are linearly independent. An easy calculation shows that the Casoratian for these functions is C(k) = 2 - 33k+3. (3.45) Since C(k) 0, we again conclude that these three functions are linearly independent.

3.2.2 Example B Consider the three functions fi (k) = 3, f2(k) = k3, f3 (k) = k23k. (3.42) The linear combinations of these three functions can be written as follows: cifi(k)+ c2f2(k) + c3f3(k) = 3*(c1 + c2k + c3k?). (3.43) Note that the right-hand side of equation (3.43) will be zero if and only if C1 + c2k + c3k2 = 0. (3.44) %3D But this can only occur for cı = c2 = c3 = 0. Therefore, the three functions given in equation (3.42) are linearly independent. An easy calculation shows that the Casoratian for these functions is C(k) = 2 - 33k+3. (3.45) Since C(k) 0, we again conclude that these three functions are linearly independent.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Explain the determaine

3.2.2
Example B
Consider the three functions
f1(k) = 3*, f2(k) = k3*, f3(k) = k²3*.
(3.42)
The linear combinations of these three functions can be written as follows:
f1(k) + c2f2(k)+ c3f3(k) = 3* (c1 + c2k + c3k²).
(3.43)
Note that the right-hand side of equation (3.43) will be zero if and only if
Co
C1 + c2k + c3k² = 0.
(3.44)
But this
an only occur for c1 = C2 = c3 = 0. Therefore, the three functions
given in equation (3.42) are linearly independent.
An easy calculation shows that the Casoratian for these functions is
C(k) = 2 - 33k+3.
(3.45)
Since C(k) 0, we again conclude that these three functions are linearly
independent.

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