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Chapter 1: Q23E (page 1)
In Exercises 23-26, describe the possible echelon forms of the matrix. Use the notation of Example 1 in Section 1.2
23. \(A\) is a \(3 \times 3\) matrix with linearly independent columns.
The possible echelon form of the \(3 \times 3\) matrix is .
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In Exercises 21 and 22, find a parametric equation of the line \(M\) through \({\mathop{\rm p}\nolimits} \) and \({\mathop{\rm q}\nolimits} \). [Hint: \(M\) is parallel to the vector \({\mathop{\rm q}\nolimits} - p\). See the figure below.]
22. \({\mathop{\rm p}\nolimits} = \left[ {\begin{array}{*{20}{c}}{ - 6}\\3\end{array}} \right]\), \(q = \left[ {\begin{array}{*{20}{c}}0\\{ - 4}\end{array}} \right]\)
Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section 1.2.
a. A is a \({\bf{2}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{2}}}\).
b. A is a \({\bf{3}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{3}}}\).
Consider each matrix in Exercises 5 and 6 as the augmented matrix of a linear system. State in words the next two elementary row operations that should be performed in the process of solving the system.
6. \(\left( {\begin{aligned}{*{20}{c}}1&{ - 6}&4&0&{ - 1}\\0&2&{ - 7}&0&4\\0&0&1&2&{ - 3}\\0&0&3&1&6\end{aligned}} \right)\)
In a certain region, about 6% of city鈥檚 population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2015, there were, 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where\({{\bf{x}}_{\bf{0}}}\)is the initial population in 2015. Then estimate the population in the city and in the suburbs two years later, in 2017.
Find an equation involving \(g,\,h,\)and \(k\) that makes this augmented matrix correspond to a consistent system:
\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2}&5&{ - 9}&k\end{array}} \right]\)
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