Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 7E from Chapter 1.4 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 7E
Chapter:
Problem:
In Exercises 5-8, use the definition of Ax to write the matrix equation as a vector equation, or vice versa...
Step-by-Step Solution
Step 1
Given Matrix Equations:
\[{x_1}\left[ {\begin{array}{*{20}{c}}4\\{ - 1}\\7\\{ - 4}\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\{ - 5}\\1\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}7\\{ - 8}\\0\\2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}9\\{ - 8}\\0\\{ - 7}\end{array}} \right]\]We have to convert the given vector form into matrix form ($A{\bf{x}} = b$). We will use coefficients of variables to find the coefficient vector (A) and constant terms as corresponding constant vector (b).
Step 2: The coefficient Matrix
\[A = \left[ {\begin{array}{*{20}{c}}4&{ - 5}&7\\{ - 1}&3&{ - 8}\\7&{ - 5}&0\\{ - 4}&1&2\end{array}} \right]\]
Step 3: The Variable Vector
\[{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right]\]
Step 4: The Constant Vector
Using the corresponding constants: \[{\bf{b}} = \left[ {\begin{array}{*{20}{c}}6\\{ - 8}\\0\\{ - 7}\end{array}} \right]\]
Step 4: ANSWERS
The Matrix Form\[\left[ {\begin{array}{*{20}{c}}4&{ - 5}&7\\{ - 1}&3&{ - 8}\\7&{ - 5}&0\\{ - 4}&1&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}6\\{ - 8}\\0\\{ - 7}\end{array}} \right]\]
Given Matrix Equations:
\[{x_1}\left[ {\begin{array}{*{20}{c}}4\\{ - 1}\\7\\{ - 4}\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\{ - 5}\\1\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}7\\{ - 8}\\0\\2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}9\\{ - 8}\\0\\{ - 7}\end{array}} \right]\]We have to convert the given vector form into matrix form ($A{\bf{x}} = b$). We will use coefficients of variables to find the coefficient vector (A) and constant terms as corresponding constant vector (b).
Step 2: The coefficient Matrix
\[A = \left[ {\begin{array}{*{20}{c}}4&{ - 5}&7\\{ - 1}&3&{ - 8}\\7&{ - 5}&0\\{ - 4}&1&2\end{array}} \right]\]
Step 3: The Variable Vector
\[{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right]\]
Step 4: The Constant Vector
Using the corresponding constants: \[{\bf{b}} = \left[ {\begin{array}{*{20}{c}}6\\{ - 8}\\0\\{ - 7}\end{array}} \right]\]
Step 4: ANSWERS
The Matrix Form\[\left[ {\begin{array}{*{20}{c}}4&{ - 5}&7\\{ - 1}&3&{ - 8}\\7&{ - 5}&0\\{ - 4}&1&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\\{{x_4}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}6\\{ - 8}\\0\\{ - 7}\end{array}} \right]\]