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The set \(B = \left\{ {1 + {t^2},t + {t^2},1 + 2t + {t^2}} \right\}\) is a basis for \({{\mathop{\rm P}\nolimits} _2}\). Find the coordinate vector of \({\mathop{\rm p}\nolimits} \left( t \right) = 1 + 4t + 7{t^2}\) relative to \(B\).

Answer:

The coordinate vector of \({\mathop{\rm p}\nolimits} \left( t \right) = 1 + 4t + 7{t^2}\) relative to \(B\) is \({\left[ {\mathop{\rm p}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}2\\6\\{ - 1}\end{array}} \right]\).

Short Answer

Expert verified

The coordinate vector of \({\mathop{\rm p}\nolimits} \left( t \right) = 1 + 4t + 7{t^2}\) relative to \(B\) is \({\left[ {\mathop{\rm p}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}2\\6\\{ - 1}\end{array}} \right]\).

Step by step solution

01

Equate the coefficients of the two polynomials

Determine \({c_1},{c_2}\), and \({c_3}\) as shown below:

\[{c_1}\left( {1 + {t^2}} \right) + {c_2}\left( {t + {t^2}} \right) + {c_3}\left( {1 + 2t + {t^2}} \right) = {\mathop{\rm p}\nolimits} \left( t \right) = 1 + 4t + 7{t^2}\]

Equate the coefficients of the two polynomials to obtain the system of equations, as shown below:

\[\begin{array}{c}{c_1} + \,\,\,\,\,\,\,{c_3} = 1\\\,\,\,\,\,\,\,\,{c_2} + 2{c_3} = 4\\{c_1} + {c_2} + {c_3} = 7\end{array}\]

02

Convert the system of equations into an augmented matrix

Convert the system of equations into an augmented matrix, as shown below:

\(\left[ {\begin{array}{*{20}{c}}1&0&1&1\\0&1&2&4\\1&1&1&7\end{array}} \right]\)

03

Apply the row operation

At row 3, subtract row 1 from row 3.

\( \sim \left[ {\begin{array}{*{20}{c}}1&0&1&1\\0&1&2&4\\0&1&0&6\end{array}} \right]\)

At row 3, subtract row 2 from row 3.

\( \sim \left[ {\begin{array}{*{20}{c}}1&0&1&1\\0&1&2&4\\0&0&{ - 2}&2\end{array}} \right]\)

At row 3, multiply row 3 by \( - \frac{1}{2}\).

\( \sim \left[ {\begin{array}{*{20}{c}}1&0&1&1\\0&1&2&4\\0&0&1&{ - 1}\end{array}} \right]\)

At row 1, subtract row 3 from row 1. At row 2, multiply row 3 by 2 and subtract it from row 2.

\( \sim \left[ {\begin{array}{*{20}{c}}1&0&0&2\\0&1&0&6\\0&0&1&{ - 1}\end{array}} \right]\)

04

Determine the coordinate vector of \({\mathop{\rm p}\nolimits} \left( t \right) = 1 + 4t + 7{t^2}\)

Therefore, \({\left[ {\mathop{\rm p}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}2\\6\\{ - 1}\end{array}} \right]\).

It is also possible to solve this problem using the coordinate vectors of the given polynomials relative to the standard basis \(\left\{ {1,t,{t^2}} \right\}\). The same system of equations will be obtained in the result.

Thus, the coordinate vector of \({\mathop{\rm p}\nolimits} \left( t \right) = 1 + 4t + 7{t^2}\) relative to \(B\) is \({\left[ {\mathop{\rm p}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}2\\6\\{ - 1}\end{array}} \right]\).

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Most popular questions from this chapter

Consider the polynomials , and \({p_{\bf{3}}}\left( t \right) = {\bf{2}}\) \({p_{\bf{1}}}\left( t \right) = {\bf{1}} + t,{p_{\bf{2}}}\left( t \right) = {\bf{1}} - t\)(for all t). By inspection, write a linear dependence relation among \({p_{\bf{1}}},{p_{\bf{2}}},\) and \({p_{\bf{3}}}\). Then find a basis for Span\(\left\{ {{p_{\bf{1}}},{p_{\bf{2}}},{p_{\bf{3}}}} \right\}\).

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.

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

Question 11: Let\(S\)be a finite minimal spanning set of a vector space\(V\). That is,\(S\)has the property that if a vector is removed from\(S\), then the new set will no longer span\(V\). Prove that\(S\)must be a basis for\(V\).

Let be a linear transformation from a vector space \(V\) \(T:V \to W\)in to vector space \(W\). Prove that the range of T is a subspace of . (Hint: Typical elements of the range have the form \(T\left( {\mathop{\rm x}\nolimits} \right)\) and \(T\left( {\mathop{\rm w}\nolimits} \right)\) for some \({\mathop{\rm x}\nolimits} ,\,{\mathop{\rm w}\nolimits} \)in \(V\).)\(W\)

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