Verify that 1k, (-2)k, 3k are linearly independent signals.

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Suresh Chand · Accepted Answer

Solution: We have, uk = 1k, vk = (-2)k, wk = 3k By castrati matrix we have (begin{bmatrix}u_k & v_k & w_k u_{k+1} & v_{k+1} & w_{k+1} u_{k+1} & v_{k+1} & w_{k+2}end{bmatrix}) = (begin{bmatrix}1^k & (-2)^k & 3^k 1^{k+1} & (-2)^{k+1} & 3^{k+1} 1^{k+2} & (-2)^{k+2} & 3^{k+2}end{bmatrix}) taking k = 0, we have, (begin{bmatrix}1 & 1 & 1 1 & -2 & 3 1 & 4 & 9end{bmatrix}) Apply R2 → R2 - R1 and R3 → R3 - R1 ~ (begin{bmatrix}1 & 1 & 1 0 & -3 & 2 0 & 3 & 8end{bmatrix}) Apply R3 → R3 + R2 ~ (begin{bmatrix}1 & 1 & 1 0 & -3 & 2 0 & 0 & 10end{bmatrix}) Since, Each column is a pivot column which shows that the castrati is invertible for k = 0, 1k, (-2)k, 3k are linearly independent.

Verify that 1^k, (-2)^k, 3^k are linearly independent signals.

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Verify that 1k, (-2)k, 3k are linearly independent signals.

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Verify that 1^k, (-2)^k, 3^k are linearly independent signals.