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Chapter 1: Q25E (page 1)

In Exercises 23-26, describe the possible echelon forms of the matrix. Use the notation of Example 1 in Section 1.2

25. \(A\) is a \(4 \times 2\) matrix, \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\end{array}} \right]\) and \({a_2}\) is not a multiple of \({a_1}\).

Short Answer

Expert verified

The echelon form of the \(4 \times 2\) matrix is ,.

Step by step solution

01

Recall the notation of example 1 used for matrices in the echelon form

In example 1, the following matrices are in echelon form. The leading entries may have any non-zero value, and the starred entries \(\left( * \right)\) may have any value (including zero).

02

Use the above notation to determine the echelon forms of the matrix

It is given that \(A\) is a \(4 \times 2\) matrix, \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\end{array}} \right]\), and \({a_2}\) is not a multiple of \({a_1}\).

Use the leading entries and starred entries \(\left( * \right)\) to construct the echelon form of the \(4 \times 2\) matrix.

,

Thus, the echelon form of the \(4 \times 2\) matrix is ,.

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Most popular questions from this chapter

Determine whether the statements that follow are true or false, and justify your answer.

18: [111315171921][-13-1]=[131921]

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.

In a grid of wires, the temperature at exterior mesh points is maintained at constant values, (in°C)as shown in the accompanying figure. When the grid is in thermal equilibrium, the temperature Tat each interior mesh point is the average of the temperatures at the four adjacent points. For example,

T2=T3+T1+200+04

Find the temperatures T1,T2,andT3andwhen the grid is in thermal equilibrium.

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.

30.\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&{ - 2}&6\\0&{ - 5}&9\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&1&{ - 3}\\0&{ - 5}&9\end{array}} \right]\)

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
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