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Chapter 4: Q12E (page 191)

If the null space of a\({\bf{5}} \times {\bf{6}}\)matrix A is 4-dimensional, what is the dimension of the row space of A?

Short Answer

Expert verified

The dimension of the row space of A is 2.

Step by step solution

01

Describe the given data

Given that \({\rm{dim}}\,{\rm{Nul}}\,A = 4\).

02

Use the rank theorem

Bythe rank theorem, you get

\(\begin{aligned}{c}{\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,A &= n\\{\rm{rank}}\,A + 4 &= 6\\{\rm{rank}}\,A &= 6 - 4\\ &= 2.\end{aligned}\)

03

Draw a conclusion

Note that \({\rm{rank}}\,A = \dim {\rm{Col}}A = \dim {\rm{Row}}A\). Hence, \(\dim {\rm{Row}}\,{\rm{A}} = 2\).

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Most popular questions from this chapter

Find a basis for the set of vectors in\({\mathbb{R}^{\bf{2}}}\)on the line\(y = {\bf{5}}x\).

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

If the null space of an \({\bf{8}} \times {\bf{5}}\) matrix A is 2-dimensional, what is the dimension of the row space of A?

Exercises 23-26 concern a vector space V, a basis \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\)and the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\).

Given vectors, \({u_{\bf{1}}}\),鈥�.,\({u_p}\) and w in V, show that w is a linear combination of \({u_{\bf{1}}}\),鈥�.,\({u_p}\) if and only if \({\left( w \right)_B}\) is a linear combination of vectors \({\left( {{{\bf{u}}_{\bf{1}}}} \right)_B}\),鈥�.,\({\left( {{{\bf{u}}_p}} \right)_B}\).

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.)
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