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Chapter 1: Q11Q (page 1)

Construct a \(2 \times 3\) matrix \(A\), not in echelon form, such that the solution of \(Ax = 0\) is a line in \({\mathbb{R}^3}\).

Short Answer

Expert verified

The matrix that is not in the echelon form is \(\left( {\begin{aligned}{*{20}{c}}1&2&1\\1&5&2\end{aligned}} \right)\).

Step by step solution

01

Determine the condition for the linear solution of \(Ax = 0\) 

A solution set is a line that has one free variable in the system. In a coefficient matrix of \(2 \times 3\), two columns should be pivot columns.

02

Construct matrix \(A\) that is not in the echelon form

Take the example \(\left( {\begin{aligned}{*{20}{c}}1&2& * \\0&3& * \end{aligned}} \right)\). Fill any value in the entries of column 3 to obtain in an echelon matrix. Perform one row replacement operation on the second row to construct a matrix that is not in the echelon form.

\(\left( {\begin{aligned}{*{20}{c}}1&2&1\\0&3&1\end{aligned}} \right) \sim \left( {\begin{aligned}{*{20}{c}}1&2&1\\1&5&2\end{aligned}} \right)\)

Thus, the matrix which is not in the echelon form is \(\left( {\begin{aligned}{*{20}{c}}1&2&1\\1&5&2\end{aligned}} \right)\).

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Most popular questions from this chapter

In Exercises 10, write a vector equation that is equivalent tothe given system of equations.

10. \(4{x_1} + {x_2} + 3{x_3} = 9\)

\(\begin{array}{c}{x_1} - 7{x_2} - 2{x_3} = 2\\8{x_1} + 6{x_2} - 5{x_3} = 15\end{array}\)

Let \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 4}\\0&3&{ - 2}\\{ - 2}&6&3\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}4\\1\\{ - 4}\end{array}} \right]\). Denote the columns of \(A\) by \({{\mathop{\rm a}\nolimits} _1},{a_2},{a_3}\) and let \(W = {\mathop{\rm Span}\nolimits} \left\{ {{a_1},{a_2},{a_3}} \right\}\).

  1. Is \(b\) in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)? How many vectors are in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)?
  2. Is \(b\) in \(W\)? How many vectors are in W.
  3. Show that \({a_1}\) is in W.[Hint: Row operations are unnecessary.]

In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers \(a,....,f\) ? Justify your answers. (Hint: Use a theorem for (b).)

  1. \(\left( {\begin{aligned}{*{20}{c}}a\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\d\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\end{aligned}} \right)\)
  2. \(\left( {\begin{aligned}{*{20}{c}}a\\1\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\1\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\\1\end{aligned}} \right)\)

Determine h and k such that the solution set of the system (i) is empty, (ii) contains a unique solution, and (iii) contains infinitely many solutions.

a. \({x_1} + 3{x_2} = k\)

\(4{x_1} + h{x_2} = 8\)

b. \( - 2{x_1} + h{x_2} = 1\)

\(6{x_1} + k{x_2} = - 2\)

In Exercises 33 and 34, Tis a linear transformation from \({\mathbb{R}^2}\) into \({\mathbb{R}^2}\). Show that T is invertible and find a formula for \({T^{ - 1}}\).

34. \(T\left( {{x_1},{x_2}} \right) = \left( {6{x_1} - 8{x_2}, - 5{x_1} + 7{x_2}} \right)\)
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