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Chapter 1: Q37Q (page 1)

In Exercises 37鈥�40, let T be the linear transformation whose standard matrix is given. In Exercises 37 and 38, decide if T is a one-to-one mapping. In Exercises 39 and 40, decide if T maps \({\mathbb{R}^{\bf{5}}}\) onto \({\mathbb{R}^{\bf{5}}}\). Justify your answers.

37. \(\left[ {\begin{array}{*{20}{c}}{ - 5}&{10}&{ - 5}&4\\8&3&{ - 4}&7\\4&{ - 9}&5&{ - 3}\\{ - 3}&{ - 2}&5&4\end{array}} \right]\)

Short Answer

Expert verified

Transformation T is not one-to-one.

Step by step solution

01

Identify the condition for one-to-one mapping

The transformation maps\({\mathbb{R}^n}\)one-to-one\({\mathbb{R}^m}\)if at most one solution exists for\(T\left( {\bf{x}} \right) = {\bf{b}}\), and each vector b is in the codomain \({\mathbb{R}^m}\). And this linear transformation, \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\), has only a trivial solution.

02

Convert the matrix into the row-reduced echelon form

Consider the matrix\(A = \left[ {\begin{array}{*{20}{c}}{ - 5}&{10}&{ - 5}&4\\8&3&{ - 4}&7\\4&{ - 9}&5&{ - 3}\\{ - 3}&{ - 2}&5&4\end{array}} \right]\).

Use the code in the MATLAB to obtain the row-reduced echelon form, as shown below:

\(\begin{array}{l} > > {\rm{ A }} = {\rm{ }}\left[ { - 5{\rm{ }}10{\rm{ }} - 5{\rm{ }}4;{\rm{ }}8{\rm{ }}3{\rm{ }} - 4{\rm{ }}7;{\rm{ }}4{\rm{ }} - 9{\rm{ }}5{\rm{ }} - 3;{\rm{ }} - 3{\rm{ }} - 2{\rm{ }}5{\rm{ }}4} \right];\\ > > {\rm{ U}} = {\rm{rref}}\left( {\rm{A}} \right)\end{array}\)

\(\left[ {\begin{array}{*{20}{c}}{ - 5}&{10}&{ - 5}&4\\8&3&{ - 4}&7\\4&{ - 9}&5&{ - 3}\\{ - 3}&{ - 2}&5&4\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&0&0&{44/35}\\0&1&0&{79/35}\\0&0&1&{86/35}\\0&0&0&0\end{array}} \right]\)

You can also write it as \(\left[ {\begin{array}{*{20}{c}}1&0&0&{1.2571}\\0&1&0&{2.2571}\\0&0&1&{2.4571}\\0&0&0&0\end{array}} \right]\).

In the obtained matrix, the fourth column does not have a pivot position. So, the solution is non-trivial.

Thus, transformation T is not one-to-one.

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Most popular questions from this chapter

Determine by inspection whether the vectors in Exercises 15-20 are linearly independent. Justify each answer.

16. \(\left[ {\begin{array}{*{20}{c}}4\\{ - 2}\\6\end{array}} \right],\left[ {\begin{array}{*{20}{c}}6\\{ - 3}\\9\end{array}} \right]\)

Consider the problem of determining whether the following system of equations is consistent:

\(\begin{aligned}{c}{\bf{4}}{x_1} - {\bf{2}}{x_2} + {\bf{7}}{x_3} = - {\bf{5}}\\{\bf{8}}{x_1} - {\bf{3}}{x_2} + {\bf{10}}{x_3} = - {\bf{3}}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of linear combinations. Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase 鈥渃olumns of A.鈥�
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let \({T_1},...,{T_4}\) denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes鈥攖o the left, above, to the right, and below. For instance,

\({T_1} = \left( {10 + 20 + {T_2} + {T_4}} \right)/4\), or \(4{T_1} - {T_2} - {T_4} = 30\)

33. Write a system of four equations whose solution gives estimates

for the temperatures \({T_1},...,{T_4}\).

In Exercises 21 and 22, mark each statement True or False. Justify each answer on the basis of a careful reading of the text.

21.

a. The columns of a matrix \(A\) are linearly independent if the equation \(Ax = 0\) has a trivial solution.

b. If \(S\) is a linearly dependent set, then each vector is a linear combination of the other vectors in \(S\).

c. The columns of any \(4 \times 5\) matrix are linearly dependent.

d. If \({\mathop{\rm x}\nolimits} \) and \(y\) are linearly independent, and if \(\left\{ {x,y,z} \right\}\) is linearly dependent, then \(z\) is in Span\(\left\{ {x,y} \right\}\).

In Exercises 15 and 16, fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables

\(\left[ {\begin{array}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3{x_1} - 2{x_3}}\\{4{x_1}}\\{{x_1} - {x_2} + {x_3}}\end{array}} \right]\)
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