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Chapter 4: Q34E (page 191)

In Exercises 33 and 34, determine whether the sets of polynomials form a basis for \({{\bf{P}}_3}\). Justify your conclusions.

(M) \({\bf{5}} - {\bf{3}}t + {\bf{4}}{t^{\bf{2}}} + {\bf{2}}{t^{\bf{3}}}\), \({\bf{9}} + t + {\bf{8}}{t^{\bf{2}}} - {\bf{6}}{t^{\bf{3}}}\), \({\bf{6}} - {\bf{2}}t + {\bf{5}}{t^{\bf{2}}}\), \({t^{\bf{3}}}\)

Short Answer

Expert verified

Set S is not the basis for \({{\bf{P}}_3}\).

Step by step solution

01

Write the polynomial in the standard vector form

\(\left\{ {5 - 3t + 4{t^2} + 2{t^3},\;9 + t + 8{t^2} - 6{t^3},6 - 2\;t + 5{t^2},\;{t^3}} \right\} = \left\{ {\left( {\begin{array}{*{20}{c}}5\\{ - 3}\\4\\2\end{array}} \right),\;\;\left( {\begin{array}{*{20}{c}}9\\1\\8\\{ - 6}\end{array}} \right),\;\;\left( {\begin{array}{*{20}{c}}6\\{ - 2}\\5\\0\end{array}} \right),\;\;\left( {\begin{array}{*{20}{c}}0\\0\\0\\1\end{array}} \right)} \right\}\)

02

Form the matrix using the vectors

The matrix formed by using the vectors of the polynomials is:

\(A = \left( {\begin{array}{*{20}{c}}5&9&6&0\\{ - 3}&1&{ - 2}&0\\4&8&5&0\\2&{ - 6}&0&1\end{array}} \right)\)

03

Write the matrix in the echelon form

Consider matrix\(A = \left( {\begin{array}{*{20}{c}}5&9&6&0\\{ - 3}&1&{ - 2}&0\\4&8&5&0\\2&{ - 6}&0&1\end{array}} \right)\).

Use code in the MATLAB to obtain the row-reducedechelon formas shown below:

\(\begin{array}{l} > > {\rm{ A }} = {\rm{ }}\left( {{\rm{ }}\begin{array}{*{20}{c}}5&9&6&{0;\;\;\begin{array}{*{20}{c}}{ - 3}&1&{ - 2}&{0;\;\;\begin{array}{*{20}{c}}4&8&5&{0;\;\;\begin{array}{*{20}{c}}2&{ - 6}&0&1\end{array}}\end{array}}\end{array}}\end{array}} \right);\\ > > {\rm{ U}} = {\rm{rref}}\left( {\rm{A}} \right)\end{array}\)

\(\left( {\begin{array}{*{20}{c}}5&9&6&0\\{ - 3}&1&{ - 2}&0\\4&8&5&0\\2&{ - 6}&0&1\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&{\frac{3}{4}}&0\\0&1&{\frac{1}{4}}&0\\0&0&0&1\\0&0&0&0\end{array}} \right)\)

It can be observed from the echelon form that the last row is zero, which shows the set is linearly dependent.

Therefore, set S is not the basis for \({{\bf{P}}_3}\).

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

Given \(T:V \to W\) as in Exercise 35, and given a subspace \(Z\) of \(W\), let \(U\) be the set of all \({\mathop{\rm x}\nolimits} \) in \(V\) such that \(T\left( {\mathop{\rm x}\nolimits} \right)\) is in \(Z\). Show that \(U\) is a subspace of \(V\).

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 \({\left( {\bf{x}} \right)_B}\)).

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

19. \(A = \left( {\begin{array}{*{20}{c}}{.9}&1&0\\0&{ - .9}&0\\0&0&{.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}0\\1\\1\end{array}} \right)\).

In Exercise 3, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

3. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{4}}}\\{\bf{3}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{2}}\\{ - {\bf{2}}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{7}}}\\{\bf{0}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{0}}\\{ - {\bf{1}}}\end{array}} \right)\)
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