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Q28E

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Linear Algebra and its Applications
Found in: Page 191
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

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

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

The polynomials are linearly independent.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Write the polynomials in the standard vector form

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

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

Step 2: Form the matrix using the vectors

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

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

Step 3: Write the matrix in the echelon form

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

From the echelon form, it can be observed that there are no free variables.

So, the polynomials are linearly independent.

Most popular questions for Math Textbooks

If the null space of A \({\bf{7}} \times {\bf{6}}\) matrix A is 4-dimensional, what is the dimension of the column space of A?

Let \(A\) be any \(2 \times 3\) matrix such that \({\mathop{\rm rank}\nolimits} A = 1\), let u be the first column of \(A\), and suppose \({\mathop{\rm u}\nolimits} \ne 0\). Explain why there is a vector v in \({\mathbb{R}^3}\) such that \(A = {{\mathop{\rm uv}\nolimits} ^T}\). How could this construction be modified if the first column of \(A\) were zero?

(M) Determine whether w is in the column space of \(A\), the null space of \(A\), or both, where

\({\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}1\\2\\1\\0\end{array}} \right),A = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0\\{ - 5}&2&1&{ - 2}\\{10}&{ - 8}&6&{ - 3}\\3&{ - 2}&1&0\end{array}} \right)\)

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

Let H be an n-dimensional subspace of an n-dimensional vector space V. Explain why \(H = V\).
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