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Chapter 4: Q15E (page 191)

If A is a \({\bf{6}} \times {\bf{8}}\) matrix, what is the smallest possible dimension of Null A?

Short Answer

Expert verified

The smallest possible dimension of Null A is 2.

Step by step solution

01

Describe the given data

From the given \(6 \times 8\) matrix, the number of pivotscannot exceed 6. That is

\({\rm{rank}}\,A \le 6\).

02

Use the rank theorem

By the rank theorem, you get

\(\begin{aligned} n &= {\rm{rank}}\,A + \dim \,{\rm{Null}}\,A\\8 &\le 6 + \dim \,{\rm{Null}}\,A\\8 - 6 &\le \dim \,{\rm{Null}}\,A\\2 &\le \dim \,{\rm{Null}}\,A.\end{aligned}\)

03

Draw a conclusion

Hence, the smallest possible dimension of Null A is 2.

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Most popular questions from this chapter

(M) Show thatis a linearly independent set of functions defined on. Use the method of Exercise 37. (This result will be needed in Exercise 34 in Section 4.5.)

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

20. \(A = \left( {\begin{array}{*{20}{c}}{.8}&{ - .3}&0\\{.2}&{.5}&1\\0&0&{ - .5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\1\\0\end{array}} \right)\).

Let \(B = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{4}}}\end{array}} \right),\,\left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{9}}\end{array}} \right)\,} \right\}\). Since the coordinate mapping determined by B is a linear transformation from \({\mathbb{R}^{\bf{2}}}\) into \({\mathbb{R}^{\bf{2}}}\), this mapping must be implemented by some \({\bf{2}} \times {\bf{2}}\) matrix A. Find it. (Hint: Multiplication by A should transform a vector x into its coordinate vector \({\left( {\bf{x}} \right)_B}\)).

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

21. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 2}&{ - 4.2}&{ - 4.8}&{ - 3.6}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

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’ right sides to make the new system inconsistent? Explain.
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