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Chapter 4: Q18Q (page 191)

Determine the dimensions of and \({\mathop{\rm Col}\nolimits} A\) for the matrices shown in Exercise 13-18

18. \(A = \left( {\begin{array}{*{20}{c}}1&4&{ - 1}\\0&7&0\\0&0&0\end{array}} \right)\)

Short Answer

Expert verified

The dimension of \({\mathop{\rm Col}\nolimits} A\) is 2, and the dimension of \({\mathop{\rm Nul}\nolimits} A\) is 1.

Step by step solution

01

State the condition for the dimensions of \({\mathop{\rm Nul}\nolimits} A\) and \({\mathop{\rm Col}\nolimits} A\)

Thedimension of \({\mathop{\rm Nul}\nolimits} A\) is thenumber of free variablesin the equation \(A{\mathop{\rm x}\nolimits} = 0\), and the dimension of \({\mathop{\rm Col}\nolimits} A\)is thenumber of pivot columnsin \(A\).

02

Determine the dimensions of \({\mathop{\rm Nul}\nolimits} A\) and \({\mathop{\rm Col}\nolimits} A\)

The given matrix A is in echelon form. As it has two pivot columns, the dimension of \({\mathop{\rm Col}\nolimits} A\) is 2. The equation \(A{\mathop{\rm x}\nolimits} = 0\) has one free variable since the matrix has one column without a pivot. Therefore, the dimension of \({\mathop{\rm Nul}\nolimits} A\) is 1.

Thus, the dimension of \({\mathop{\rm Col}\nolimits} A\) is 2 and of \({\mathop{\rm Nul}\nolimits} A\) is 1.

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Most popular questions from this chapter

Suppose \({{\bf{p}}_{\bf{1}}}\), \({{\bf{p}}_{\bf{2}}}\), \({{\bf{p}}_{\bf{3}}}\), and \({{\bf{p}}_{\bf{4}}}\) are specific polynomials that span a two-dimensional subspace H of \({P_{\bf{5}}}\). Describe how one can find a basis for H by examining the four polynomials and making almost no computations.

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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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