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Chapter 4: Q19E (page 191)

Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.

Short Answer

Expert verified

Yes, by using the rank theorem and the invertible matrix theorem.

Step by step solution

01

Describe the given statement

Consider the homogeneous system \(Ax = 0\), where A is the \(5 \times 6\) matrix. From the given statement, \({\rm{dim Null }}A = 1\).

02

Use the rank theorem

By the rank theorem, you get

\(\begin{aligned} {\rm{rank }}A &= n - {\rm{dim Null }}A\\ &= 6 - 1\\{\rm{rank }}A &= 5.\end{aligned}\)

As \({\rm{dim Col }}A = {\rm{rank }}A\), \({\rm{dim Col }}A = 5\). Since Col A is the subspace of \({\mathbb{R}^5}\), \({\rm{Col }}A = {\mathbb{R}^5}\).

03

Draw a conclusion

By the invertible matrix theorem, for every b in \({\mathbb{R}^5}\), the system \(Ax = b\) has a unique solution.

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Most popular questions from this chapter

In Exercises 19 and 20, \(V\) is a vector space. Mark each statement True or False. Justify each answer.

19.

a. The number of pivot columns of a matrix equals the dimension of its column space.

b. A plane in \({\mathbb{R}^3}\) is a two-dimensional subspace of \({\mathbb{R}^3}\).

c. The dimension of the vector space \({{\mathop{\rm P}\nolimits} _4}\) is 4.

d. If \(\dim V = n\) and \(S\) is a linearly independent set in \(V\), then \(S\) is a basis for \(V\).

e. If a set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) spans a finite-dimensional vector space \(V\) and if \(T\) is a set of more than p vectors in \(V\), then \(T\) is linearly dependent.

Determine the dimensions of \({\mathop{\rm Nul}\nolimits} A\) and \({\mathop{\rm Col}\nolimits} A\) for the matrices shown in Exercise 13-18

16. \(A = \left( {\begin{array}{*{20}{c}}3&4\\{ - 6}&{10}\end{array}} \right)\)

A homogeneous system of twelve linear equations in eight unknowns has two fixed solutions that are not multiples of each other, and all other solutions are linear combinations of these two solutions. Can the set of all solutions be described with fewer than twelve homogeneous linear equations? If so, how many? Discuss.

In Exercises 29 and 30, V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. Mark each statement True or False. Justify each answer. (These questions are some difficult than those 19 and 20.)

30.

a. If there exists a linearly dependent set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) in \(V\), then \(\dim V \le p\).

b. If every set of \(p\) elements in \(V\) fails to span \(V\), then \(\dim V > p\).

c. If \(p \ge 2\), and \(\dim V = p\), then every set of \(p - 1\) nonzero vectors is linearly independent.

Qis any matrix such that

\({\left( {\bf{v}} \right)_C} = Q{\left( {\bf{v}} \right)_B}\)for each v in V (9)

Set \({\bf{v}} = {{\bf{b}}_{\bf{1}}}\) in (9). Then (9) shows that \({\left( {{{\bf{b}}_{\bf{1}}}} \right)_C}\) is the first column of Q because (a) _____. Similarly, for \(k = {\bf{2}}\),鈥�..n the kth column of Q is (b) _____ because (c) _____. This shows the matrix \(\mathop P\limits_{C \leftarrow B} \) defined by (5) in Theorem 15 is the only matrix that satisfies condition (4).
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