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Chapter 4: Q20E (page 191)
Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations鈥� right sides to make the new system inconsistent? Explain.
No, by using the rank theorem and the invertible matrix theorem.
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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.
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.
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 Exercise 4, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).
4. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\\{\bf{0}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\\{\bf{2}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{7}}}\\{\bf{3}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{8}}\\{ - {\bf{7}}}\end{array}} \right)\)
If A is a \({\bf{7}} \times {\bf{5}}\) matrix, what is the largest possible rank of A? If Ais a \({\bf{5}} \times {\bf{7}}\) matrix, what is the largest possible rank of A? Explain your answer.
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