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Chapter 4: Q37E (page 191)

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

\({\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}1\\1\\{ - 1}\\{ - 3}\end{array}} \right),A = \left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1\\{ - 5}&{ - 1}&0&{ - 2}\\9&{ - 11}&7&{ - 3}\\{19}&{ - 9}&7&1\end{array}} \right)\)

Short Answer

Expert verified

\({\mathop{\rm w}\nolimits} \)is inCol A, and w is not in Nul A.

Step by step solution

01

Write an augmented matrix

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

\(\left( {\begin{array}{*{20}{c}}A&w\end{array}} \right) = \left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1&1\\{ - 5}&{ - 1}&0&{ - 2}&1\\9&{ - 11}&7&{ - 3}&{ - 1}\\{19}&{ - 9}&7&1&{ - 3}\end{array}} \right)\)

02

Convert the matrix into row-reduced echelon form

Consider the matrix \(A = \left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1&1\\{ - 5}&{ - 1}&0&{ - 2}&1\\9&{ - 11}&7&{ - 3}&{ - 1}\\{19}&{ - 9}&7&1&{ - 3}\end{array}} \right)\).

Use the following code in MATLAB to obtain the row-reduced echelon form.

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( {7\,\,\,6\,\,\, - 4\,\,\,1\,\,\,1;\,\, - 5\,\,\, - 1\,\,\,0\,\,\, - 2\,\,\,1;\,\,9\,\,\, - 11\,\,\,7\,\,\, - 3\,\,\, - 1;\,19\,\,\, - 9\,\,\,7\,\,\,1\,\,\, - 3} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm rref}\nolimits} \left( {\mathop{\rm A}\nolimits} \right)\end{array}\)

\(\left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1&1\\{ - 5}&{ - 1}&0&{ - 2}&1\\9&{ - 11}&7&{ - 3}&{ - 1}\\{19}&{ - 9}&7&1&{ - 3}\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&0&{\frac{{ - 1}}{{95}}}&{\frac{1}{{95}}}\\0&1&0&{\frac{{39}}{{19}}}&{\frac{{ - 20}}{{19}}}\\0&0&1&{\frac{{267}}{{95}}}&{\frac{{ - 172}}{{95}}}\\0&0&0&0&0\end{array}} \right)\)

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

\(\begin{array}{l} > > {\mathop{\rm A}\nolimits} = \left( {7\,\,\,6\,\,\, - 4\,\,\,1;\,\, - 5\,\,\, - 1\,\,\,0\,\,\, - 2;\,\,9\,\,\, - 11\,\,\,7\,\,\, - 3;\,19\,\,\, - 9\,\,\,7\,\,\,1} \right)\\ > > {\mathop{\rm w}\nolimits} = \left( {1;\,\,1;\,\, - 1;\,\, - 3} \right)\\ > > {\mathop{\rm U}\nolimits} = {\mathop{\rm A}\nolimits} * w\end{array}\)

\(\begin{array}{c}{\mathop{\rm A}\nolimits} {\mathop{\rm w}\nolimits} = \left( {\begin{array}{*{20}{c}}7&6&{ - 4}&1\\{ - 5}&{ - 1}&0&{ - 2}\\9&{ - 11}&7&{ - 3}\\{19}&{ - 9}&7&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\1\\{ - 1}\\{ - 3}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{14}\\0\\0\\0\end{array}} \right)\end{array}\)

Thus, w is not in Nul A.

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Most popular questions from this chapter

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

22. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 1}&{ - 13}&{ - 12.2}&{ - 1.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

Given \(T:V \to W\) as in Exercise 35, and given a subspace \(Z\) of \(W\), let \(U\) be the set of all \({\mathop{\rm x}\nolimits} \) in \(V\) such that \(T\left( {\mathop{\rm x}\nolimits} \right)\) is in \(Z\). Show that \(U\) is a subspace of \(V\).

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 a subset \(\left\{ {{{\bf{u}}_1},...,{{\bf{u}}_p}} \right\}\) in V is linearly independent if and only if the set of coordinate vectors \(\left\{ {{{\left( {{{\bf{u}}_{\bf{1}}}} \right)}_B},.....,{{\left( {{{\bf{u}}_p}} \right)}_B}} \right\}\) is linearly independent in \({\mathbb{R}^n}\)(Hint: Since the coordinate mapping is one-to-one, the following equations have the same solutions, \({c_{\bf{1}}}\),鈥�.,\({c_p}\))

\({c_{\bf{1}}}{{\bf{u}}_{\bf{1}}} + ..... + {c_p}{{\bf{u}}_p} = {\bf{0}}\) The zero vector V

\({\left( {{c_{\bf{1}}}{{\bf{u}}_{\bf{1}}} + ..... + {c_p}{{\bf{u}}_p}} \right)_B} = {\left( {\bf{0}} \right)_B}\) The zero vector in \({\mathbb{R}^n}\)a

In Exercise 17, Ais an \(m \times n\] matrix. Mark each statement True or False. Justify each answer.

17. a. The row space of A is the same as the column space of \({A^T}\].

b. If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.

c. The dimensions of the row space and the column space of A are the same, even if Ais not square.

d. The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

e. On a computer, row operations can change the apparent rank of a matrix.

In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.
See all solutions

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