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Chapter 1: Q7E (page 1)

In Exercises 7-12, describe all solutions of \(Ax = 0\) in parametric vector form, where \(A\) is row equivalent to the given matrix.

7. \(\left[ {\begin{array}{*{20}{c}}1&3&{ - 3}&7\\0&1&{ - 4}&5\end{array}} \right]\)

Short Answer

Expert verified

The general solution in the parametric vector form is \(x = {x_3}\left[ {\begin{array}{*{20}{c}}{ - 9}\\4\\1\\0\end{array}} \right] + {x_4}\left[ {\begin{array}{*{20}{c}}8\\{ - 5}\\0\\1\end{array}} \right]\).

Step by step solution

01

Write the matrix as an augmented matrix

The augmented matrix \(\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right]\) for the given matrix is represented as:

\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 3}&7&0\\0&1&{ - 4}&5&0\end{array}} \right]\)

02

Apply row operation

Perform an elementary row operation to produce the first augmented matrix.

Perform the sum of \( - 3\) times row 2 and row 1 at row 1.

\(\left[ {\begin{array}{*{20}{c}}1&0&9&{ - 8}&0\\0&1&{ - 4}&5&0\end{array}} \right]\)

03

Convert the matrix into the equation

To obtain the solution of the system of equations, you have to convert the augmented matrix into the system of equations again.

Write the obtained matrix \(\left[ {\begin{array}{*{20}{c}}1&0&9&{ - 8}&0\\0&1&{ - 4}&5&0\end{array}} \right]\)into the equation notation.

\[\begin{array}{c}{x_1} + 9{x_3} - 8{x_4} = 0\\{x_2} - 4{x_3} + 5{x_4} = 0\end{array}\]

04

Determine the basic variable and free variable of the system

The variables corresponding to the pivot columns in the matrix are calledbasic variables.The other variable is called a free variable.

\({x_1}\)and \({x_2}\) are basic variables, and \({x_3}\) and \({x_4}\) are free variables.

Thus, \({x_1} = - 9{x_3} + 8{x_4},{x_2} = 4{x_3} - 5{x_4}\).

05

Determine the general solution in the parametric vector form

Sometimes the parametric form of an equation is written as\(x = s{\mathop{\rm u}\nolimits} + t{\mathop{\rm v}\nolimits} \,\,\,\left( {s,t\,{\mathop{\rm in}\nolimits} \,\mathbb{R}} \right)\).

The general solution of \(Ax = 0\) in the parametric vector form can be represented as:

\(\begin{array}{c}x = \left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ - 9{x_3} + 8{x_4}}\\{4{x_3} - 5{x_4}}\\{{x_3}}\\{{x_4}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ - 9{x_3}}\\{4{x_3}}\\{{x_3}}\\0\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{8{x_4}}\\{ - 5{x_4}}\\0\\{{x_4}}\end{array}} \right]\\ = {x_3}\left[ {\begin{array}{*{20}{c}}{ - 9}\\4\\1\\0\end{array}} \right] + {x_4}\left[ {\begin{array}{*{20}{c}}8\\{ - 5}\\0\\1\end{array}} \right]\end{array}\)

Thus, the general solution in the parametric vector form is \(x = {x_3}\left[ {\begin{array}{*{20}{c}}{ - 9}\\4\\1\\0\end{array}} \right] + {x_4}\left[ {\begin{array}{*{20}{c}}8\\{ - 5}\\0\\1\end{array}} \right]\).

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Most popular questions from this chapter

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 \(U\left( {\mathop{\rm v}\nolimits} \right)\)).

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

16: There exists a 2x2 matrix such that

A[11]=[12]andA[22]=[21].

Question: There exists a 2x2 matrix such thatA[12]=[34].

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?

Find the general solutions of the systems whose augmented matrices are given as

12. \(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\{ - 1}&7&{ - 4}&2&7\end{array}} \right]\).
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