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Chapter 4: Q16E (page 191)

If A is a \({\bf{6}} \times {\bf{4}}\) matrix, what is the smallest possible dimension of Null A?

Short Answer

Expert verified

The smallest possible dimension of Null A is 0.

Step by step solution

01

Describe the given data

From the given \(6 \times 4\) matrix, the number of pivots cannot exceed 4. That is

\({\rm{rank}}\,A \le 4\).

02

Use the rank theorem

Bythe rank theorem, you get

\(\begin{aligned} n &= {\rm{rank}}\,A + \dim \,{\rm{Null}}\,A\\4 &\le 4 + \dim \,{\rm{Null}}\,A\\4 - 4 &\le \dim \,{\rm{Null}}\,A\\0 &\le \dim \,{\rm{Null}}\,A.\end{aligned}\)

03

Draw a conclusion

The smallest possible dimension of Null Ais 0.

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Most popular questions from this chapter

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

16. If \(A\) is an \(m \times n\) matrix of rank\(r\), then a rank factorization of \(A\) is an equation of the form \(A = CR\), where \(C\) is an \(m \times r\) matrix of rank\(r\) and \(R\) is an \(r \times n\) matrix of rank \(r\). Such a factorization always exists (Exercise 38 in Section 4.6). Given any two \(m \times n\) matrices \(A\) and \(B\), use rank factorizations of \(A\) and \(B\) to prove that rank\(\left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

(Hint: Write \(A + B\) as the product of two partitioned matrices.)

Let \(A\) be an \(m \times n\) matrix of rank \(r > 0\) and let \(U\) be an echelon form of \(A\). Explain why there exists an invertible matrix \(E\) such that \(A = EU\), and use this factorization to write \(A\) as the sum of \(r\) rank 1 matrices. [Hint: See Theorem 10 in Section 2.4.]

The null space of a \({\bf{5}} \times {\bf{6}}\) matrix A is 4-dimensional, what is the dimension of the column space of A.

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.

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

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