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Q14E

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Linear Algebra and its Applications
Found in: Page 1
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

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

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

The general solution of the system is

\({x_1} = - 5 + 2{x_2} + 5{x_3} + 6{x_4}\)

\({x_2} = 2 + 6{x_3} + 3{x_4}\)

\({x_3}\) is free.

\({x_4}\)is free.

\({x_5} = 0\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Convert the given matrix into the equation

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

Write the given matrix into the equation notation.

\(\begin{aligned}{c}{x_1} - 2{x_2} - 5{x_3} - 6{x_4} = - 5\\{x_2} - 6{x_3} - 3{x_4} = 2\\{x_5} = 0\end{aligned}\)

Step 2: Determine the basic variable and free variable

The variables corresponding to the pivot columns in the matrix are called basic variables.

The other variables are called free variables.

The basic variables of the given matrix are \({x_1},{x_2},{x_5}\). The free variables are \({x_3},{x_4}\).

Step 3: Write the general solution of the system

Thus, the general solution of the system is

\(\begin{aligned}{c}{x_1} = - 5 + 2{x_2} + 5{x_3} + 6{x_4}\\{x_2} = 2 + 6{x_3} + 3{x_4}\end{aligned}\)

\({x_3}\) is a free variable.

\({x_4}\) is a free variable.

\({x_5} = 0\).

Most popular questions for Math Textbooks

Find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

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

In Exercises 13 and 14, determine if \(b\) is a linear combination of the vectors formed from the columns of the matrix \(A\).

13. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 4}&2\\0&3&5\\{ - 2}&8&{ - 4}\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}3\\{ - 7}\\{ - 3}\end{array}} \right]\)

In Exercise 19 and 20, choose \(h\) and \(k\) such that the system has

a. no solution

b. unique solution

c. many solutions.

Give separate answers for each part.

19. \(\begin{array}{l}{x_1} + h{x_2} = 2\\4{x_1} + 8{x_2} = k\end{array}\)

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

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

Let T be a linear transformation that maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\). Is \({T^{ - 1}}\) also one-to-one?

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