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In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work

\({\left( {{\bf{1}} - t} \right)^{\bf{2}}}\),\(t - {\bf{2}}{t^{\bf{2}}} + {t^{\bf{3}}}\),\({\left( {{\bf{1}} - t} \right)^{\bf{3}}}\)

Short Answer

Expert verified

The polynomials are linearly dependent.

Step by step solution

01

Write the polynomials in the standard vector form

The vectors of the given polynomials can be written as follows:

\(\begin{array}{c}{\left( {1 - t} \right)^2} = 1 - 2t + {t^2}\\ \equiv \left( {\begin{array}{*{20}{c}}1\\{ - 2}\\1\\0\end{array}} \right)\end{array}\),

\(t - 2{t^2} + {t^3} \equiv \left( {\begin{array}{*{20}{c}}0\\1\\{ - 2}\\1\end{array}} \right)\)

and

\(\begin{array}{c}{\left( {1 - t} \right)^3} = 1 - 3t + 3{t^2} - {t^3}\\ \equiv \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\3\\{ - 1}\end{array}} \right)\end{array}\)

02

Form the matrix using the vectors

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

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

03

Write the matrix in the echelon form

\(\left( {\begin{array}{*{20}{c}}1&0&1\\{ - 2}&1&{ - 3}\\1&{ - 2}&3\\0&1&{ - 1}\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&1\\0&1&{ - 1}\\0&0&0\\0&0&0\end{array}} \right)\)

From the echelon form, it can be observed that for three variables, there are two equations. Hence, one free variable is present.

So, the polynomials are linearly dependent.

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Most popular questions from this chapter

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\(V\).

(M) Show thatis a linearly independent set of functions defined on. Use the method of Exercise 37. (This result will be needed in Exercise 34 in Section 4.5.)

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 \(V\).

In Exercise 2, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

2. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{5}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{7}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{8}}\\{ - {\bf{5}}}\end{array}} \right)\)

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