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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 18: Suppose A is a \(4 \times 4\) matrix and B is a \(4 \times 2\) matrix, and let \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) represent a sequence of input vectors in \({\mathbb{R}^2}\).

  1. Set \({{\mathop{\rm x}\nolimits} _0} = 0\), compute \({{\mathop{\rm x}\nolimits} _1},...,{{\mathop{\rm x}\nolimits} _4}\) from equation (1), and write a formula for \({{\mathop{\rm x}\nolimits} _4}\) involving the controllability matrix \(M\) appearing in equation (2). (Note: The matrix \(M\) is constructed as a partitioned matrix. Its overall size here is \(4 \times 8\).)
  2. Suppose \(\left( {A,B} \right)\) is controllable and v is any vector in \({\mathbb{R}^4}\). Explain why there exists a control sequence \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) in \({\mathbb{R}^2}\) such that \({{\mathop{\rm x}\nolimits} _4} = {\mathop{\rm v}\nolimits} \).

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

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

\({\left( {{\bf{1}} - t} \right)^{\bf{2}}}\),\(t - {\bf{2}}{t^{\bf{2}}} + {t^{\bf{3}}}\),\({\left( {{\bf{1}} - t} \right)^{\bf{3}}}\)

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

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

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

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