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In Exercises 7-12, assume the signals listed are solutions of the given difference equation. Determine if the signals form a basis for the solution space of the equation. Justify your answers using appropriate theorems.

1k, 2k, (2)k, yk+3yk+24yk+1+4yk=0

Short Answer

Expert verified

The dimension of H is 3, so the three linearly independent signals form a basis of H.

Step by step solution

01

Write the Casorati matrix

The Casorati matrix of the solution is

Ak=[1k2k(2)k1k+12k+1(2)k+11k+22k+2(2)k+2].

02

Check the Casorati matrix for k=0

Substitute 0 for k in the Casorati matrix.

A0=[1020(2)010+120+1(2)0+110+220+2(2)0+2]=[111122144][100010001]

The Casorati matrix is row equivalent to theidentity matrix. Therefore, it is invertible.

Hence, the set of signals {1k,2k,(2)k} is linearly independent. The dimension of H is 3, so the three linearly independent signals form a basis of H.

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Most popular questions from this chapter

Let S be a maximal linearly independent subset of a vector space V. In other words, S has the property that if a vector not in S is adjoined to S, the new set will no longer be linearly independent. Prove that S must be a basis of V. [Hint: What if S were linearly independent but not a basis of V?]

If A is a 4×3 matrix, what is the largest possible dimension of the row space of A? If Ais a 3×4 matrix, what is the largest possible dimension of the row space of A? Explain.

Use coordinate vector to test whether the following sets of poynomial span P2. Justify your conclusions.

a. 13t+5t2, 3+5t7t2, 4+5t6t2, 1t2

b. 5t+t2, 18t2t2, 3+4t+2t2, 23t

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is m×n.

17. A submatrix of a matrix A is any matrix that results from deleting some (or no) rows and/or columns of A. It can be shown that A has rank r if and only if A contains an invertible r×r submatrix and no longer square submatrix is invertible. Demonstrate part of this statement by explaining (a) why an m×n matrix A of rank r has an m×r submatrix A1 of rank r, and (b) why A1 has an invertible r×r submatrix A2.

The concept of rank plays an important role in the design of engineering control systems, such as the space shuttle system mentioned in this chapter’s introductory example. A state-space model of a control system includes a difference equation of the form

xk+1=Axk+Bukfor k=0,1,.... (1)

Where A is n×n, B is n×m, {xk} is a sequence of “state vectors” in Rn that describe the state of the system at discrete times, and {uk} is a control, or input, sequence. The pair (A,B) is said to be controllable if

rank(BABA2BAn1B)=n (2)

The matrix that appears in (2) is called the controllability matrix for the system. If (A,B) is controllable, then the system can be controlled, or driven from the state 0 to any specified state v (in Rn) in at most n steps, simply by choosing an appropriate control sequence in Rm. This fact is illustrated in Exercise 18 for n=4 and m=2. For a further discussion of controllability, see this text’s website (Case study for Chapter 4).

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

20. A=(.8.30.2.5100.5),B=(110).

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