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Chapter 4: Q21E (page 191)

Let \(B = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{4}}}\end{array}} \right),\,\left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{9}}\end{array}} \right)\,} \right\}\). Since the coordinate mapping determined by B is a linear transformation from \({\mathbb{R}^{\bf{2}}}\) into \({\mathbb{R}^{\bf{2}}}\), this mapping must be implemented by some \({\bf{2}} \times {\bf{2}}\) matrix A. Find it. (Hint: Multiplication by A should transform a vector x into its coordinate vector \({\left( {\bf{x}} \right)_B}\)).

Short Answer

Expert verified

\(\left( {\begin{array}{*{20}{c}}9&2\\4&1\end{array}} \right)\)

Step by step solution

01

Find the value of x

The coordinate of x relative to B can be expressed as:

\(\begin{array}{c}{\bf{x}} = {c_1}\left( {\begin{array}{*{20}{c}}1\\{ - 4}\end{array}} \right) + {c_2}\left( {\begin{array}{*{20}{c}}{ - 2}\\9\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}1&{ - 2}\\{ - 4}&9\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{c_1}}\\{{c_2}}\end{array}} \right)\\ = {P_B}{\left( {\bf{x}} \right)_B}\end{array}\)

02

Find matrix A

Matrix A is the inverse of \({P_B}\).

\(\begin{array}{l}A = P_B^{ - 1}\\ = \left| {{P_B}} \right| \cdot {\rm{Adj}}\left( {{P_B}} \right)\\ = \left| {\begin{array}{*{20}{c}}1&{ - 2}\\{ - 4}&9\end{array}} \right| \cdot \left( {\begin{array}{*{20}{c}}9&2\\4&1\end{array}} \right)\\ = \left( {9 - 8} \right)\left( {\begin{array}{*{20}{c}}9&2\\4&1\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}9&2\\4&1\end{array}} \right)\end{array}\)

So, matrix A is \(\left( {\begin{array}{*{20}{c}}9&2\\4&1\end{array}} \right)\).

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Most popular questions from this chapter

In statistical theory, a common requirement is that a matrix be of full rank. That is, the rank should be as large as possible. Explain why an m n matrix with more rows than columns has full rank if and only if its columns are linearly independent.

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 \({{\mathop{\rm x}\nolimits} _4} = {\mathop{\rm v}\nolimits} \).

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

21. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 2}&{ - 4.2}&{ - 4.8}&{ - 3.6}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

Use Exercise 28 to explain why the equation\(Ax = b\)has a solution for all\({\rm{b}}\)in\({\mathbb{R}^m}\)if and only if the equation\({A^T}x = 0\)has only the trivial solution.

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

20. \(A = \left( {\begin{array}{*{20}{c}}{.8}&{ - .3}&0\\{.2}&{.5}&1\\0&0&{ - .5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\1\\0\end{array}} \right)\).
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