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Chapter 1: Q28E (page 1)
How many pivot columns must a \(5 \times 7\) matrix have if its columns span \({\mathbb{R}^5}\)? Why?
The \(5 \times 7\) matrix must have five pivot columns.
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In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a de铿乶ition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.
23.
a. Every elementary row operation is reversible.
b. A \(5 \times 6\)matrix has six rows.
c. The solution set of a linear system involving variables \({x_1},\,{x_2},\,{x_3},........,{x_n}\)is a list of numbers \(\left( {{s_1},\, {s_2},\,{s_3},........,{s_n}} \right)\) that makes each equation in the system a true statement when the values \ ({s_1},\, {s_2},\, {s_3},........,{s_n}\) are substituted for \({x_1},\,{x_2},\,{x_3},........,{x_n}\), respectively.
d. Two fundamental questions about a linear system involve existence and uniqueness.
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?
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 \({\mathbb{R}^{\bf{3}}}\).
In Exercises 6, write a system of equations that is equivalent to the given vector equation.
6. \({x_1}\left[ {\begin{array}{*{20}{c}}{ - 2}\\3\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}8\\5\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}1\\{ - 6}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\end{array}} \right]\)
Construct three different augmented matrices for linear systems whose solution set is \({x_1} = - 2,{x_2} = 1,{x_3} = 0\).
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