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Chapter 1: Q6Q (page 1)

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.

Short Answer

Expert verified

a. Vectorb is in the span \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\).

b. Vector b is in linear combinations of the columns of matrix A.

c. Vector b is in the range of T.

Step by step solution

01

(a) Step 1: Write the system in the vector form

The system of equations\(4{x_1} - 2{x_2} + 7{x_3} = - 5\)and\(8{x_1} - 3{x_2} + 10{x_3} = - 3\)can be represented in the matrix equation form \(A{\bf{x}} = {\bf{b}}\) as follows:

\(\left( {\begin{aligned}{*{20}{c}}4&{ - 2}&7\\8&{ - 3}&{10}\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{ - 5}\\{ - 3}\end{aligned}} \right)\)

Assume that the column vectors are \({{\bf{v}}_1} = \left( {\begin{aligned}{*{20}{c}}4\\8\end{aligned}} \right)\), \({{\bf{v}}_2} = \left( {\begin{aligned}{*{20}{c}}{ - 2}\\{ - 3}\end{aligned}} \right)\), \({{\bf{v}}_3} = \left( {\begin{aligned}{*{20}{c}}7\\{10}\end{aligned}} \right)\), and \({\bf{b}} = \left( {\begin{aligned}{*{20}{c}}{ - 5}\\{ - 3}\end{aligned}} \right)\).

02

Convert the system of equations into the row-reduced echelon form

Convert the system of equations\(4{x_1} - 2{x_2} + 7{x_3} = - 5\)and\(8{x_1} - 3{x_2} + 10{x_3} = - 3\)into the augmented matrix as shown below:

\(\left( {\begin{aligned}{*{20}{c}}4&{ - 2}&7&{ - 5}\\8&{ - 3}&{10}&{ - 3}\end{aligned}} \right)\)

Use the \(4{x_1}\) term in the first equation to eliminate the \(8{x_1}\) term from the second equation. Add \( - 2\) times row one to row two.

\(\left( {\begin{aligned}{*{20}{c}}4&{ - 2}&7&{ - 5}\\0&1&{ - 4}&7\end{aligned}} \right)\)

In the above augmented matrix, there are no free variables. It means the system of equations is consistent.

Thus, b is in the span \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\).

03

(b) Step 3: Write in terms of columns of A

The system of equations\(4{x_1} - 2{x_2} + 7{x_3} = - 5\)and\(8{x_1} - 3{x_2} + 10{x_3} = - 3\)or the augmented matrix \(\left( {\begin{aligned}{*{20}{c}}4&{ - 2}&7&{ - 5}\\8&{ - 3}&{10}&{ - 3}\end{aligned}} \right)\) in the vector form is shown below:

\({x_1}\left( {\begin{aligned}{*{20}{c}}4\\8\end{aligned}} \right) + {x_2}\left( {\begin{aligned}{*{20}{c}}{ - 2}\\{ - 3}\end{aligned}} \right) + {x_3}\left( {\begin{aligned}{*{20}{c}}7\\{10}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{ - 5}\\{ - 3}\end{aligned}} \right)\)

From the above row-reduced echelon form,

\(\left( {\begin{aligned}{*{20}{c}}4&{ - 2}&7&{ - 5}\\0&1&{ - 4}&7\end{aligned}} \right)\).

The system of equations is consistent.

Thus, b is in linear combinations of the columns of matrix A.

04

(c) Step 4: If b is in the range of T

Consider the transformation\(T\left( {\bf{x}} \right) = A{\bf{x}}\).

It can also be written as shown below:

\(T\left( {\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right)} \right) = \left( {\begin{aligned}{*{20}{c}}4&{ - 2}&7\\8&{ - 3}&{10}\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right)\)

Let the solution be\({\bf{x}} = \left( {\begin{aligned}{*{20}{c}}{9/4}\\7\\0\end{aligned}} \right)\).

Then, the transformation becomes:

\(\begin{aligned}{c}T\left( {\left( {\begin{aligned}{*{20}{c}}{9/4}\\7\\0\end{aligned}} \right)} \right) = \left( {\begin{aligned}{*{20}{c}}4&{ - 2}&7\\8&{ - 3}&{10}\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{9/4}\\7\\0\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{4\left( {9/4} \right) - 14 + 0}\\{8\left( {9/4} \right) - 21 + 0}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{ - 5}\\{ - 3}\end{aligned}} \right)\\ = {\bf{b}}.\end{aligned}\)

So,\(T\left( {\bf{x}} \right) = {\bf{b}}\).

Thus, vector b is in the range of T.

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Most popular questions from this chapter

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation. Explain why T is both one-to-one and onto \({\mathbb{R}^n}\). Use equations (1) and (2). Then give a second explanation using one or more theorems.

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation, and let Sand U be functions from \({\mathbb{R}^n}\) into \({\mathbb{R}^n}\) such that \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) and \(\)\(U\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\). Show that \(U\left( v \right) = S\left( v \right)\) for all v in \({\mathbb{R}^n}\). This will show that Thas a unique inverse, as asserted in theorem 9. (Hint: Given any v in \({\mathbb{R}^n}\), we can write \({\mathop{\rm v}\nolimits} = T\left( {\mathop{\rm x}\nolimits} \right)\) for some x. Why? Compute \(S\left( {\mathop{\rm v}\nolimits} \right)\) and \(U\left( {\mathop{\rm v}\nolimits} \right)\)).

Solve the systems in Exercises 11鈥�14.

12.\(\begin{aligned}{c}{x_1} - 3{x_2} + 4{x_3} = - 4\\3{x_1} - 7{x_2} + 7{x_3} = - 8\\ - 4{x_1} + 6{x_2} - {x_3} = 7\end{aligned}\)

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?

Suppose \(a,b,c,\) and \(d\) are constants such that \(a\) is not zero and the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the numbers \(a,b,c,\) and \(d\)? Justify your answer.

28. \(\begin{array}{l}a{x_1} + b{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)
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