Explanation Given: The set of signals 2 k , 4 k , ( − 5 ) k and the difference equation y k + 3 − y k + 2 − 22 y k + 1 + 40 y k = 0 . Theorem used: The set H of all solutions of the n th -order homogeneous linear difference equation y k + n + a 1 y k + n + 1 + ... + a n − 1 y k + 1 + a n y k = 0 for all k is an n -dimensional vector space. Justification: Take the Casorati Matrix of the signals in order to the check the linear independency of the given signal and then check the invertible of the matrix. Write Casorati Matrix for the given solution. [ 2 k 4 k ( − 5 ) k 2 k + 1 4 k + 1 ( − 5 ) k + 1 2 k + 2 4 k + 2 ( − 5 ) k + 2 ] Compute and use row reduce the Casorati Matrix for the signals 2 k , 4 k , ( − 5 ) k . Setting k = 0 to check the resultant matrix for invertible

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Chapter 4.8, Problem 8E

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Whether the set of signals 2k,4k,(−5)k form a basis for the solution space of the difference equation and justify through the theorem.

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Chapter 4.8, Problem 7E

Chapter 4.8, Problem 9E

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Chapter 4 Solutions

EBK LINEAR ALGEBRA AND ITS APPLICATIONS

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Whether the set of signals 2 k , 4 k , ( − 5 ) k form a basis for the solution space of the difference equation and justify through the theorem.

Solution Summary: The author analyzes the Casorati Matrix of the signals to determine the solution space for the n th -order homogeneous linear difference equation.

Whether the set of signals 2k,4k,(−5)k form a basis for the solution space of the difference equation and justify through the theorem.

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Chapter 4 Solutions