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Q24E

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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

Is it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side? Explain.

Yes, it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants.

No, it is not possible for such a system to have a unique solution for every right-hand side.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Describe the given statement

It is given that a nonhomogeneous system has seven linear equations with six unknowns. The system has unique solutions for some right-hand side of constants. This implies that the system has at most six pivot positions.

Step 2: Use the rank theorem

Consider the nonhomogeneous system \(Ax = b\), where \(A\) is \(7 \times 6\) matrix. As the system has at most six pivot positions, \({\rm{rank}}\,A \le 6\), and the value of unknown’s \(n\) is 6 . By the rank theorem, \({\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A = n\).

Put the values as shown:

\(\begin{aligned} {\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A &= n\\{\rm{dim}}\,{\rm{Nul}}\,\,A &= n - {\rm{rank}}\,A\\{\rm{dim}}\,{\rm{Nul}}\,\,A &\ge 6 - 6\\{\rm{dim}}\,{\rm{Nul}}\,\,A &\ge 0\end{aligned}\)

Step 3: Draw a conclusion

If \({\rm{dim}}\,{\rm{Nul}}\,\,A = 0\), the system \(Ax = b\) has no free variable and its solution is unique. The value of \({\rm{dimcol}}\,A\) is also 6. Moreover, \({\rm{col}}\,A\) is a subspace of \({\mathbb{R}^7}\) as \({\rm{rank}}\,A \le 6\). So, a value of \(b\) must exist in \({\mathbb{R}^7}\)at which the nonhomogeneous system \(Ax = b\) is inconsistent. Thus, the system \(Ax = b\) may not have a unique solution for all \(b\).

Most popular questions for Math Textbooks

If A is a \({\bf{7}} \times {\bf{5}}\) matrix, what is the largest possible rank of A? If A is a \({\bf{5}} \times {\bf{7}}\) matrix, what is the largest possible rank of A? Explain your answer.

Find a basis for the set of vectors in \({\mathbb{R}^{\bf{3}}}\) in the plane \(x + {\bf{2}}y + z = {\bf{0}}\). (Hint: Think of the equation as a “system” of homogeneous equations.)

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

15. Let \(A\) be an \(m \times n\) matrix, and let \(B\) be a \(n \times p\) matrix such that \(AB = 0\). Show that \({\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B \le n\). (Hint: One of the four subspaces \({\mathop{\rm Nul}\nolimits} A\), \({\mathop{\rm Col}\nolimits} A,\,{\mathop{\rm Nul}\nolimits} B\), and \({\mathop{\rm Col}\nolimits} B\) is contained in one of the other three subspaces.)

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\(\left\{ {{p_{\bf{1}}},{p_{\bf{2}}},{p_{\bf{3}}}} \right\}\).

Which of the subspaces \({\rm{Row }}A\) , \({\rm{Col }}A\), \({\rm{Nul }}A\), \({\rm{Row}}\,{A^T}\) , \({\rm{Col}}\,{A^T}\) , and \({\rm{Nul}}\,{A^T}\) are in \({\mathbb{R}^m}\) and which are in \({\mathbb{R}^n}\) ? How many distinct subspaces are in this list?.
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