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Linear Algebra and its Applications
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Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

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

The matrix that 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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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: 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.

Step 2: 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 \(\left( {\begin{aligned}{*{20}{c}}1&2&1\\1&5&2\end{aligned}} \right)\).

Most popular questions for Math Textbooks

Solve each system in Exercises 1–4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure.

  1. \(\begin{aligned}{c}{x_1} + 5{x_2} = 7\\ - 2{x_1} - 7{x_2} = - 5\end{aligned}\)

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

Consider a dynamical system x→(t+1)=Ax→(t) with two components. The accompanying sketch shows the initial state vector x→0 and two eigen vectors υ1→  and  υ2→ of A (with eigen values λ1→ and λ2→ respectively). For the given values of λ1→ and λ2→ , draw a rough trajectory. Consider the future and the past of the system.

λ1→=1,λ2→=0.9

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.

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 “columns of A.”

  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.
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