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Let \(B\) be the basis of \({{\mathop{\rm P}\nolimits} _3}\) consisting of the Hermite polynomials in Exercise 21, and let \(p\left( t \right) = 7 - 12t - 8{t^2} + 12{t^3}\). Find the coordinate vector of \({\mathop{\rm p}\nolimits} \) relative to \(B\).

Short Answer

Expert verified

The coordinate vector of \({\mathop{\rm p}\nolimits} \) relative to \(B\) is \({\left[ {\mathop{\rm p}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}3\\3\\{ - 2}\\{\frac{3}{2}}\end{array}} \right]\).

Step by step solution

01

Definition of the coordinate vector of x

Suppose\(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\)is a basis for\(V\)and x is in\(V\). Thecoordinatesof\({\mathop{\rm x}\nolimits} \)relative to the basis \(B\)(or the\(B\)-coordinates of x) are the weights \({c_1},...,{c_n}\) such that \({\mathop{\rm x}\nolimits} = {c_1}{b_1} + ... + {c_n}{b_n}\).

02

Determine the coordinate vector of \({\mathop{\rm p}\nolimits} \) relative to \(B\)

The coordinate vector of\(p\left( t \right) = 7 - 12t - 8{t^2} + 12{t^3}\)with respect to\(B\)is as shown below:

\({c_1}\left( 1 \right) + {c_2}\left( {2t} \right) + {c_3}\left( { - 2 + 4{t^2}} \right) + {c_4}\left( { - 12t + 8{t^3}} \right) = 7 - 12t - 8{t^2} + 12{t^3}\)

Equate the coefficient of\(t\)to produce the system of the equation as shown below:

\(\begin{aligned} {c_1}\,\,\,\,\,\,\,\,\,\, - 2{c_3}\,\,\,\,\,\,\,\,\,\,\,\,\, &= 7\\\,\,\,\,\,\,\,2{c_2}\,\,\,\,\,\,\,\,\,\, - 12{c_4} &= - 12\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4{c_3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, &= - 8\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,8{c_4} &= 12\end{aligned}\)

By solving the system of the equation, you get\({c_1} = 3,{c_2} = 3,{c_3} = - 2,{c_4} = \frac{3}{2}\). Therefore,\({\left[ {\mathop{\rm p}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}3\\3\\{ - 2}\\{\frac{3}{2}}\end{array}} \right]\).

Thus, the coordinate vector of \({\mathop{\rm p}\nolimits} \) relative to \(B\) is \({\left[ {\mathop{\rm p}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}3\\3\\{ - 2}\\{\frac{3}{2}}\end{array}} \right]\).

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Most popular questions from this chapter

Exercises 23-26 concern a vector space V, a basis \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\) and the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\).

Show the coordinate mapping is one to one. (Hint: Suppose \({\left( {\bf{u}} \right)_B} = {\left( {\bf{w}} \right)_B}\) for some u and w in V, and show that \({\bf{u}} = {\bf{w}}\)).

In Exercise 4, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

4. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\\{\bf{0}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\\{\bf{2}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{7}}}\\{\bf{3}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{8}}\\{ - {\bf{7}}}\end{array}} \right)\)

In Exercise 1, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

1. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{6}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{3}}\end{array}} \right)\)

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

Suppose \(A\) is \(m \times n\)and \(b\) is in \({\mathbb{R}^m}\). What has to be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent?

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