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Chapter 4: Q13E (page 191)

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.

Short Answer

Expert verified

If A is a \(7 \times 5\) matrix, then the largest possible rank of A is 5.

If Ais a \(5 \times 7\) matrix, then the largest possible rank of A is 5.

Step by step solution

01

Use the rank theorem

Note that the number of pivotsin A gives the dimension of its column space. And by the rank theorem, \(\dim {\rm{Col}}\,A = \dim {\rm{Row}}\,A = {\rm{rank}}\,A\).

02

Compute the rank for \({\bf{7}} \times {\bf{5}}\) matrix

If A is a \(7 \times 5\) matrix, then the number of pivots cannot exceed the number of columns. Here, the number of columns is minimum. Hence, the largest possible rank of A is 5.

03

Compute the rank for \({\bf{5}} \times {\bf{7}}\) matrix

If A is a \(5 \times 7\) matrix, then the number of pivots cannot exceed the number of rows. Here, the number of rows is minimum. Hence, the largest possible rank of A is 5.

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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.

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.

Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution? Discuss.

Let \({y_k} = {k^2}\)and \({z_k} = 2k\left| k \right|\). Are the signals \(\left\{ {{y_k}} \right\}\) and \(\left\{ {{z_k}} \right\}\) linearly independent? Evaluate the associated Casorati matrix \(C\left( k \right)\) for \(k = 0\), \(k = - 1\), and \(k = - 2\), and discuss your results.

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

In Exercises 15 and 16, mark each statement True or False. Justify each answer. Unless stated otherwise, \(B\) is a basis for a vector space \(V\).

16.

  1. If\(B\)is the standard basis for\({\mathbb{R}^n}\)then the\(B\)-coordinate vector of an\({\mathop{\rm x}\nolimits} \)in\({\mathbb{R}^n}\)is x itself.
  2. The correspondence\({\left( {\mathop{\rm x}\nolimits} \right)_B} \mapsto {\mathop{\rm x}\nolimits} \)is called coordinate mapping.
  3. In some cases, a plane in\({\mathbb{R}^3}\)can be isomorphic to\({\mathbb{R}^2}\).
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