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Chapter 1: Q24E (page 1)
Supposea system of linear equation has a\(3 \times 5\)augmented matrix whose fifth column is a pivot column. Is the system consistent? Why (or why not)?
The \(3 \times 5\) augmented matrix is inconsistent.
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Question: There exists a 2x2 matrix such that.
In Exercise 23 and 24, make each statement True or False. Justify each answer.
23.
a. Another notation for the vector \(\left[ {\begin{array}{*{20}{c}}{ - 4}\\3\end{array}} \right]\) is \(\left[ {\begin{array}{*{20}{c}}{ - 4}&3\end{array}} \right]\).
b. The points in the plane corresponding to \(\left[ {\begin{array}{*{20}{c}}{ - 2}\\5\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 5}\\2\end{array}} \right]\) lie on a line through the origin.
c. An example of a linear combination of vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) is the vector \(\frac{1}{2}{{\mathop{\rm v}\nolimits} _1}\).
d. The solution set of the linear system whose augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}&b\end{array}} \right]\) is the same as the solution set of the equation\({{\mathop{\rm x}\nolimits} _1}{a_1} + {x_2}{a_2} + {x_3}{a_3} = b\).
e. The set Span \(\left\{ {u,v} \right\}\) is always visualized as a plane through the origin.
Write the reduced echelon form of a \(3 \times 3\) matrix A such that the first two columns of Aare pivot columns and
\(A = \left( {\begin{aligned}{*{20}{c}}3\\{ - 2}\\1\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}0\\0\\0\end{aligned}} \right)\).
Use the accompanying figure to write each vector listed in Exercises 7 and 8 as a linear combination of u and v. Is every vector in \({\mathbb{R}^2}\) a linear combination of u and v?
8.Vectors w, x, y, and z
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 definition 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.
24.
a. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
b. Two matrices are row equivalent if they have the same number of rows.
c. An inconsistent system has more than one solution.
d. Two linear systems are equivalent if they have the same solution set.
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