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Chapter 1: Q26E (page 1)

Suppose \(Ax = b\) has a solution. Explain why the solution is unique precisely when \(Ax = 0\) has only the trivial solution.

Short Answer

Expert verified

The solution is unique if and only if the corresponding system of equations has no free variables, and each column of \(A\) is a pivot column. It can only happen if the equation \(A{\mathop{\rm x}\nolimits} = 0\) has only a trivial solution.

Step by step solution

01

Use the geometric argument based on theorem 6

Theorem 6 states that equation \(A{\mathop{\rm x}\nolimits} = b\) is consistent, and the solution set of \(A{\mathop{\rm x}\nolimits} = b\) can be obtained by translating the solution set of \(Ax = 0\) using any particular solution \(p\) of \(A{\mathop{\rm x}\nolimits} = b\) for the translation.

Thus, the solution of \(A{\mathop{\rm x}\nolimits} = b\) has a single vector if and only if its solution set \(Ax = 0\) is a single vector. This occurs only when \(Ax = 0\) has only a trivial solution.

02

Use the free variables to demonstrate the theorem

It is known that the homogeneous equation \(Ax = 0\) has a nontrivial solutionif and only if the equation has at least one free variable.

When \(Ax = b\) has a solution, it will be unique if and only if the corresponding system of equations has no free variables, and each column of \(A\) is a pivot column. It can only happen if the equation \(A{\mathop{\rm x}\nolimits} = 0\) has only a trivial solution.

Thus, the solution is unique if and only if the corresponding system of equations has no free variables, and each column of \(A\) is a pivot column. It can only happen if the equation \(A{\mathop{\rm x}\nolimits} = 0\) has only a trivial solution.

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Most popular questions from this chapter

Let \({{\bf{a}}_1}\) \({{\bf{a}}_2}\), and b be the vectors in \({\mathbb{R}^{\bf{2}}}\) shown in the figure, and let \(A = \left( {\begin{aligned}{*{20}{c}}{{{\bf{a}}_1}}&{{{\bf{a}}_2}}\end{aligned}} \right)\). Does the equation \(A{\bf{x}} = {\bf{b}}\) have a solution? If so, is the solution unique? Explain.

In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer.(If true, give the approximate location where a similar statement appears, or refer to a definition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.

24.

a. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.

b. Two matrices are row equivalent if they have the same number of rows.

c. An inconsistent system has more than one solution.

d. Two linear systems are equivalent if they have the same solution set.

Suppose an experiment leads to the following system of equations:

\(\begin{aligned}{c}{\bf{4}}.{\bf{5}}{x_{\bf{1}}} + {\bf{3}}.{\bf{1}}{x_{\bf{2}}} = {\bf{19}}.{\bf{249}}\\1.6{x_{\bf{1}}} + 1.1{x_{\bf{2}}} = 6.843\end{aligned}\) (3)

  1. Solve system (3), and then solve system (4), below, in which the data on the right have been rounded to two decimal places. In each case, find the exactsolution.

\(\begin{aligned}{c}{\bf{4}}.{\bf{5}}{x_{\bf{1}}} + {\bf{3}}.{\bf{1}}{x_{\bf{2}}} = {\bf{19}}.{\bf{25}}\\1.6{x_{\bf{1}}} + 1.1{x_{\bf{2}}} = 6.8{\bf{4}}\end{aligned}\) (4)

  1. The entries in (4) differ from those in (3) by less than .05%. Find the percentage error when using the solution of (4) as an approximation for the solution of (3).
  1. Use your matrix program to produce the condition number of the coefficient matrix in (3).

In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers \(a,....,f\) ? Justify your answers. (Hint: Use a theorem for (b).)

  1. \(\left( {\begin{aligned}{*{20}{c}}a\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\d\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\end{aligned}} \right)\)
  2. \(\left( {\begin{aligned}{*{20}{c}}a\\1\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\1\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\\1\end{aligned}} \right)\)

Construct a \(3 \times 3\) matrix\(A\), with nonzero entries, and a vector \(b\) in \({\mathbb{R}^3}\) such that \(b\) is not in the set spanned by the columns of\(A\).
See all solutions

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