Chapter 1: Q25E (page 1)
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
The given coefficient matrix is consistent.
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Find the general solutions of the systems whose augmented matrices are given as
14. \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&{ - 6}&0&{ - 5}\\0&1&{ - 6}&{ - 3}&0&2\\0&0&0&0&1&0\\0&0&0&0&0&0\end{array}} \right]\).
In Exercises 9, write a vector equation that is equivalent to
the given system of equations.
9. \({x_2} + 5{x_3} = 0\)
\(\begin{array}{c}4{x_1} + 6{x_2} - {x_3} = 0\\ - {x_1} + 3{x_2} - 8{x_3} = 0\end{array}\)
Find the general solutions of the systems whose augmented matrices are given as
12. \(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\{ - 1}&7&{ - 4}&2&7\end{array}} \right]\).
Suppose \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2}} \right\}\) is a linearly independent set in \({\mathbb{R}^n}\). Show that \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2}} \right\}\) is also linearly independent.
Use Theorem 7 in section 1.7 to explain why the columns of the matrix Aare linearly independent.
\(A = \left( {\begin{aligned}{*{20}{c}}1&0&0&0\\2&5&0&0\\3&6&8&0\\4&7&9&{10}\end{aligned}} \right)\)
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