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Chapter 4: Q1E (page 191)

Verify that the signals in Exercises 1 and 2 are solutions of the accompanying difference equation.

\({{\bf{2}}^k}\),\({\left( { - {\bf{4}}} \right)^k}\);\({y_{k + {\bf{2}}}} + {\bf{2}}{y_{k + {\bf{1}}}} - {\bf{8}}{y_k} = {\bf{0}}\)

Short Answer

Expert verified

\({2^k}\), \({\left( { - 4} \right)^k}\) are the solutions of the difference equation \({y_{k + 2}} + 2{y_{k + 1}} - 8{y_k} = 0\).

Step by step solution

01

Check the given difference equation for \({{\bf{2}}^k}\)

If\({2^k}\)is the solution,

\({y_{k + 2}} = {2^{k + 2}}\),\({y_{k + 1}} = {2^{k + 1}}\)and\({y_k} = {2^k}\).

By the difference equation, you get

\(\begin{aligned}{c}{2^{k + 2}} + 2\left( {{2^{k + 1}}} \right) - 8\left( {{2^k}} \right) &= 0\\{2^k}\left( {{2^2} + {2^2} - 8} \right) &= 0\\{2^k}\left( {8 - 8} \right) &= 0.\end{aligned}\)

So,\({2^k}\)is the solution of the given difference equation.

02

Check the given difference equation for \({\left( { - {\bf{4}}} \right)^k}\)

If\({\left( { - 4} \right)^k}\)is the solution,

\({y_{k + 2}} = {\left( { - 4} \right)^{k + 2}}\), \({y_{k + 1}} = {\left( { - 4} \right)^{k + 1}}\) and \({y_k} = {\left( { - 4} \right)^k}\).

By the difference equation, you get

\(\begin{aligned} {\left( { - 4} \right)^{k + 2}} + 2\left[ {{{\left( { - 4} \right)}^{k + 1}}} \right] - 8\left[ {{{\left( { - 4} \right)}^k}} \right] &= 0\\{\left( { - 4} \right)^k}\left( {{{\left( { - 4} \right)}^2} + 2\left( { - 4} \right) - 8} \right) &= 0\\{\left( { - 4} \right)^k}\left( {16 + 8 - 8} \right) &= 0.\end{aligned}\)

So, \({\left( { - 4} \right)^k}\) is the solution of the difference equation.

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Most popular questions from this chapter

Question 18: Suppose A is a \(4 \times 4\) matrix and B is a \(4 \times 2\) matrix, and let \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) represent a sequence of input vectors in \({\mathbb{R}^2}\).

  1. Set \({{\mathop{\rm x}\nolimits} _0} = 0\), compute \({{\mathop{\rm x}\nolimits} _1},...,{{\mathop{\rm x}\nolimits} _4}\) from equation (1), and write a formula for \({{\mathop{\rm x}\nolimits} _4}\) involving the controllability matrix \(M\) appearing in equation (2). (Note: The matrix \(M\) is constructed as a partitioned matrix. Its overall size here is \(4 \times 8\).)
  2. Suppose \(\left( {A,B} \right)\) is controllable and v is any vector in \({\mathbb{R}^4}\). Explain why there exists a control sequence \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) in \({\mathbb{R}^2}\) such that \({{\mathop{\rm x}\nolimits} _4} = {\mathop{\rm v}\nolimits} \).

The null space of a \({\bf{5}} \times {\bf{6}}\) matrix A is 4-dimensional, what is the dimension of the column space of A.

(M) Let \({{\mathop{\rm a}\nolimits} _1},...,{{\mathop{\rm a}\nolimits} _5}\) denote the columns of the matrix \(A\), where \(A = \left( {\begin{array}{*{20}{c}}5&1&2&2&0\\3&3&2&{ - 1}&{ - 12}\\8&4&4&{ - 5}&{12}\\2&1&1&0&{ - 2}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}{{{\mathop{\rm a}\nolimits} _1}}&{{{\mathop{\rm a}\nolimits} _2}}&{{{\mathop{\rm a}\nolimits} _4}}\end{array}} \right)\)

  1. Explain why \({{\mathop{\rm a}\nolimits} _3}\) and \({{\mathop{\rm a}\nolimits} _5}\) are in the column space of \(B\).
  2. Find a set of vectors that spans \({\mathop{\rm Nul}\nolimits} A\).
  3. Let \(T:{\mathbb{R}^5} \to {\mathbb{R}^4}\) be defined by \(T\left( x \right) = A{\mathop{\rm x}\nolimits} \). Explain why \(T\) is neither one-to-one nor onto.

Let \({M_{2 \times 2}}\) be the vector space of all \(2 \times 2\) matrices, and define \(T:{M_{2 \times 2}} \to {M_{2 \times 2}}\) by \(T\left( A \right) = A + {A^T}\), where \(A = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\).

  1. Show that \(T\)is a linear transformation.
  2. Let \(B\) be any element of \({M_{2 \times 2}}\) such that \({B^T} = B\). Find an \(A\) in \({M_{2 \times 2}}\) such that \(T\left( A \right) = B\).
  3. Show that the range of \(T\) is the set of \(B\) in \({M_{2 \times 2}}\) with the property that \({B^T} = B\).
  4. Describe the kernel of \(T\).

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

  1. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} A\). (Hint: Explain why every vector in the column space of \(AB\) is in the column space of \(A\).
  2. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} B\). (Hint: Use part (a) to study rank\({\left( {AB} \right)^T}\).)
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