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T1(t) x2(t) a Suppose x'(t) = Ax(t) where A = , а, b, с, d € R and z(t) — Use the d 0, definition of linear independence to justify each of the following. Recall that if v E R" then each component of v must equal 0. (a) Suppose the eigenvalues of A are distinct and real-valued. Show that the terms in the solution x(t) are linearly independent. (b) Suppose there is only one eigenvalue of A. Show that the terms in the solution x(t) are linearly independent. (Hint: There are two casees here linearly independent eigenvectors associated with that eigenvalue and the second case is when we cannot find two linearly independent eigenvalues) - one case when there are two (c) Suppose the eigenvalues of A are complex conjugates. Show that the terms in the solution x(t) are linearly independent.

T1(t) x2(t) a Suppose x'(t) = Ax(t) where A = , а, b, с, d € R and z(t) — Use the d 0, definition of linear independence to justify each of the following. Recall that if v E R" then each component of v must equal 0. (a) Suppose the eigenvalues of A are distinct and real-valued. Show that the terms in the solution x(t) are linearly independent. (b) Suppose there is only one eigenvalue of A. Show that the terms in the solution x(t) are linearly independent. (Hint: There are two casees here linearly independent eigenvectors associated with that eigenvalue and the second case is when we cannot find two linearly independent eigenvalues) - one case when there are two (c) Suppose the eigenvalues of A are complex conjugates. Show that the terms in the solution x(t) are linearly independent.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.1: Eigenvalues And Eigenvectors
Problem 8E
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X1(t) x2(t) a Suppose x'(t) = Ax(t) where A = ), a, b, c, d e R and æ(t) Use the d definition of linear independence to justify each of the following. Recall that if v E R" = 0, then each component of v must equal 0. (a) Suppose the eigenvalues of A are distinct and real-valued. Show that the terms in the solution x(t) are linearly independent. (b) Suppose there is only one eigenvalue of A. Show that the terms in the solution x(t) are linearly independent. (Hint: There are two casees here linearly independent eigenvectors associated with that eigenvalue and the second case is when we cannot find two linearly independent eigenvalues) - one case when there are two (c) Suppose the eigenvalues of A are complex conjugates. Show that the terms in the solution x(t) are linearly independent.
Transcribed Image Text:X1(t) x2(t) a Suppose x'(t) = Ax(t) where A = ), a, b, c, d e R and æ(t) Use the d definition of linear independence to justify each of the following. Recall that if v E R" = 0, then each component of v must equal 0. (a) Suppose the eigenvalues of A are distinct and real-valued. Show that the terms in the solution x(t) are linearly independent. (b) Suppose there is only one eigenvalue of A. Show that the terms in the solution x(t) are linearly independent. (Hint: There are two casees here linearly independent eigenvectors associated with that eigenvalue and the second case is when we cannot find two linearly independent eigenvalues) - one case when there are two (c) Suppose the eigenvalues of A are complex conjugates. Show that the terms in the solution x(t) are linearly independent.
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