Q6E In Exercises 5鈥�8, use the defi... [FREE SOLUTION]

91影视

Linear Algebra and its Applications

David C. Lay, Steven R. Lay and Judi J. McDonald

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 483 pages

ISBN: 978-03219822384

Chapter 1: Q6E (page 1)

In Exercises 5鈥�8, use the definition ofAx to write the matrix
equation as a vector equation, or vice versa.
6. $\left[ {\begin{array}{{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]$

Short Answer

Expert verified

The matrix equation as a vector equation is$ - 2 \cdot \left[ {\begin{array}{*{20}{c}}7\\2\\9\\{ - 3}\end{array}} \right] - 5 \cdot \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\{ - 6}\\2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]$.

Step by step solution

Write the definition of $A{\bf{x}}$

It is known that the column of matrix $A$ is represented as $\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right]$, and vector x is represented as $\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]$.

According to the definition, the weights in a linear combination of matrix A columns are represented by the entries in vector x.

The matrix equation as a vector equation can be written as shown below:

$\begin{array}{c}A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\\b = {x_1}{a_1} + {x_2}{a_2} + \cdots + {x_n}{a_n}\end{array}$

The number of columns in matrix $A$ should be equal to the number of entries in vector x so that $A{\bf{x}}$ can be defined.

Write matrix A and vector x

Consider the equation $\left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]$.

Here, $A = \left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]$, ${\bf{x}} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right]$, and $b = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]$.

Write matrix A columns and vector x entries

Also, it is observed that ${{\bf{a}}_1} = \left[ {\begin{array}{*{20}{c}}7\\2\\9\\{ - 3}\end{array}} \right]$, ${{\bf{a}}_2} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\{ - 6}\\2\end{array}} \right]$, ${x_1} = - 2$, and ${x_2} = - 5$.

Use the definition to write the matrix equation as a vector equation

Write the matrix equation $\left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]$ as a vector equationby using the definition as shown below:

$\begin{array}{c}\left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\\ - 2 \cdot \left[ {\begin{array}{*{20}{c}}7\\2\\9\\{ - 3}\end{array}} \right] - 5 \cdot \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\{ - 6}\\2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\end{array}$

Thus, the matrix equation $\left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]$can be written as a vector equation as $ - 2 \cdot \left[ {\begin{array}{*{20}{c}}7\\2\\9\\{ - 3}\end{array}} \right] - 5 \cdot \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\{ - 6}\\2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]$.