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Chapter 4: Q38E (page 191)

(M) Determine whether w is in the column space of \(A\), the null space of \(A\), or both, where

\({\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}1\\2\\1\\0\end{array}} \right),A = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0\\{ - 5}&2&1&{ - 2}\\{10}&{ - 8}&6&{ - 3}\\3&{ - 2}&1&0\end{array}} \right)\)

Short Answer

Expert verified

\({\mathop{\rm w}\nolimits} \)is inCol A, and w is in Nul A.

Step by step solution

01

Write an augmented matrix

Consider the augmented matrix \(\left( {\begin{array}{*{20}{c}}A&{\mathop{\rm w}\nolimits} \end{array}} \right)\) as shown below:

\(\left( {\begin{array}{*{20}{c}}A&w\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0&1\\{ - 5}&2&1&{ - 2}&2\\{10}&{ - 8}&6&{ - 3}&1\\3&{ - 2}&1&0&0\end{array}} \right)\)

02

Convert the matrix into row-reduced echelon form

Consider the matrix \(A = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0&1\\{ - 5}&2&1&{ - 2}&2\\{10}&{ - 8}&6&{ - 3}&1\\3&{ - 2}&1&0&0\end{array}} \right)\).

Use the code in MATLAB to obtain the row-reduced echelon form as shown below:

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( { - 8\,\,5\,\, - 2\,\,\,0\,\,\,1;\, - 5\,\,\,2\,\,\,1\,\,\, - 2\,\,\,2;\,10\,\,\, - 8\,\,\,6\,\,\, - 3\,\,\,1;\,3\,\,\, - 2\,\,\,1\,\,\,\,0\,\,\,0} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm rref}\nolimits} \left( {\mathop{\rm A}\nolimits} \right)\end{array}\)

\(\left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0&1\\{ - 5}&2&1&{ - 2}&2\\{10}&{ - 8}&6&{ - 3}&1\\3&{ - 2}&1&0&0\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&{ - 1}&0&{ - 2}\\0&1&{ - 2}&0&{ - 3}\\0&0&1&1&1\\0&0&0&0&0\end{array}} \right)\)

03

Determine whether w is in the column space of A

A typical vector v in Col A has the property that the equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm v}\nolimits} \) is consistent.

The system of the equation of matrix A is consistent.

Thus, \({\mathop{\rm w}\nolimits} \) is inCol A.

04

Determine whether w is in the Null space of A

A typical vector v in Nul A has the property that \(A{\mathop{\rm v}\nolimits} = 0\).

Use the code in the MATLAB to compute the matrix \({\mathop{\rm Aw}\nolimits} \) as shown below:

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( { - 8\,\,5\,\, - 2\,\,\,0;\, - 5\,\,\,2\,\,\,1\,\,\, - 2;\,10\,\,\, - 8\,\,\,6\,\,\, - 3;\,3\,\,\, - 2\,\,\,1\,\,\,\,0} \right)\\ > > {\mathop{\rm w}\nolimits} = \left( {1;\,\,2;\,\,1;\,\,\,0} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm A}\nolimits} * w\end{array}\)

\(\begin{array}{c}{\mathop{\rm A}\nolimits} {\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}{ - 8}&5&{ - 2}&0\\{ - 5}&2&1&{ - 2}\\{10}&{ - 8}&6&{ - 3}\\3&{ - 2}&1&0\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\2\\1\\0\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}0\\0\\0\\0\end{array}} \right)\end{array}\)

Thus, w is in Nul A.

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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.)

Suppose \(A\) is \(m \times n\)and \(b\) is in \({\mathbb{R}^m}\). What has to be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent?

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

Show that the coordinate mapping is onto \({\mathbb{R}^n}\). That is, given any y in \({\mathbb{R}^n}\), with entries \({y_{\bf{1}}}\),….,\({y_n}\), produce u in V such that \({\left( {\bf{u}} \right)_B} = y\).

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.

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

19. \(A = \left( {\begin{array}{*{20}{c}}{.9}&1&0\\0&{ - .9}&0\\0&0&{.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}0\\1\\1\end{array}} \right)\).
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