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Chapter 4: Q21E (page 191)

Suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss.

Short Answer

Expert verified

No, it is not possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other.

Step by step solution

01

Describe the given statement

It is given that a nonhomogeneous system has nine linear equations with ten unknowns. A solution exists for all the possible constants on the right side of the equations. This implies that the maximum rank of matrix formed from the nonhomogeneous system is 9 as it has 9 pivot places.

02

Use the rank theorem

The value of \(n\) of the unknown is 10, and the rank is 9. By the rank theorem,\({\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A = n\).

Put the values as shown below:

\(\begin{aligned} {\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A &= n\\9 + {\rm{dim}}\,{\rm{Nul}}\,\,A &= 10\\{\rm{dim}}\,{\rm{Nul}}\,\,A &= 10 - 9\\{\rm{dim}}\,{\rm{Nul}}\,\,A &= 1\end{aligned}\)

03

Draw a conclusion

As the value of \({\rm{dim}}\,{\rm{Nul }}A\) is 1, the number of non-pivot columns is 1. Thus, it is not possible to find two nonzero solutions of the associated homogeneous system.

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Most popular questions from this chapter

Let \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\) be a basis for a vector space \(V\). Explain why the \(B - \)coordinate vectors of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}\) are the columns \({{\mathop{\rm e}\nolimits} _1},...,{{\mathop{\rm e}\nolimits} _n}\) of the \(n \times n\) identity matrix.

Answer:

The \(B - \)coordinate vectors of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}\) are columns \({{\mathop{\rm e}\nolimits} _1},...,{{\mathop{\rm e}\nolimits} _n}\) of the \(n \times n\) identity matrix.

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