Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

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 \({{\bf{P}}_3}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Define by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right)}\end{array}} \right)\). For instance, if \({\mathop{\rm p}\nolimits} \left( t \right) = 3 + 5t + 7{t^2}\), then \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}3\\{15}\end{array}} \right)\).

  1. Show that \(T\) is a linear transformation. (Hint: For arbitrary polynomials p, q in \({{\mathop{\rm P}\nolimits} _2}\), compute \(T\left( {{\mathop{\rm p}\nolimits} + {\mathop{\rm q}\nolimits} } \right)\) and \(T\left( {c{\mathop{\rm p}\nolimits} } \right)\).)
  2. Find a polynomial p in \({{\mathop{\rm P}\nolimits} _2}\) that spans the kernel of \(T\), and describe the range of \(T\).

Find a basis for the set of vectors in\({\mathbb{R}^{\bf{2}}}\)on the line\(y = {\bf{5}}x\).

In Exercise 18, Ais an \(m \times n\) matrix. Mark each statement True or False. Justify each answer.

18. a. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.

b. Row operations preserve the linear dependence relations among the rows of A.

c. The dimension of the null space of A is the number of columns of A that are not pivot columns.

d. The row space of \({A^T}\) is the same as the column space of A.

e. If A and B are row equivalent, then their row spaces are the same.

Define a linear transformation by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 0 \right)}\end{array}} \right)\). Find \(T:{{\mathop{\rm P}\nolimits} _2} \to {\mathbb{R}^2}\)polynomials \({{\mathop{\rm p}\nolimits} _1}\) and \({{\mathop{\rm p}\nolimits} _2}\) in \({{\mathop{\rm P}\nolimits} _2}\) that span the kernel of T, and describe the range of T.

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.

a. What is the dimension of range of T if T is one-to-one mapping? Explain.

b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free