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Chapter 1: Q11E (page 1)
Determine which of the matrices in Exercises 7鈥�12areorthogonal. If orthogonal, find the inverse.
11. \(\left( {\begin{aligned}{{}}{2/3}&{2/3}&{1/3}\\0&{1/3}&{ - 2/3}\\{5/3}&{ - 4/3}&{ - 2/3}\end{aligned}} \right)\)
The given matrix is not an orthogonal matrix.
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In Exercise 23 and 24, make each statement True or False. Justify each answer.
24.
a. Any list of five real numbers is a vector in \({\mathbb{R}^5}\).
b. The vector \({\mathop{\rm u}\nolimits} \) results when a vector \({\mathop{\rm u}\nolimits} - v\) is added to the vector \({\mathop{\rm v}\nolimits} \).
c. The weights \({{\mathop{\rm c}\nolimits} _1},...,{c_p}\) in a linear combination \({c_1}{v_1} + \cdot \cdot \cdot + {c_p}{v_p}\) cannot all be zero.
d. When are \({\mathop{\rm u}\nolimits} \) nonzero vectors, Span \(\left\{ {u,v} \right\}\) contains the line through \({\mathop{\rm u}\nolimits} \) and the origin.
e. Asking whether the linear system corresponding to an augmented matrix \(\left[ {\begin{array}{*{20}{c}}{{{\rm{a}}_{\rm{1}}}}&{{{\rm{a}}_{\rm{2}}}}&{{{\rm{a}}_{\rm{3}}}}&{\rm{b}}\end{array}} \right]\) has a solution amounts to asking whether \({\mathop{\rm b}\nolimits} \) is in Span\(\left\{ {{a_1},{a_2},{a_3}} \right\}\).
Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.
18. \(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&{ - 2}\\5&h&{ - 7}\end{array}} \right]\)
Suppose a linear transformation \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) has the property that \(T\left( {\mathop{\rm u}\nolimits} \right) = T\left( {\mathop{\rm v}\nolimits} \right)\) for some pair of distinct vectors u and v in \({\mathbb{R}^n}\). Can Tmap \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\)? Why or why not?
Let \(A\) be a \(3 \times 3\) matrix with the property that the linear transformation \({\bf{x}} \mapsto A{\bf{x}}\) maps \({\mathbb{R}^3}\) into \({\mathbb{R}^3}\). Explain why transformation must be one-to-one.
Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.
29. \(\left[ {\begin{array}{*{20}{c}}0&{ - 2}&5\\1&4&{ - 7}\\3&{ - 1}&6\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&4&{ - 7}\\0&{ - 2}&5\\3&{ - 1}&6\end{array}} \right]\)
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