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Chapter 1: Q5E (page 1)

In Exercises 5鈥�8, use the definition ofAx to write the matrix

equation as a vector equation, or vice versa.

5. \(\left[ {\begin{array}{*{20}{c}}5&1&{ - 8}&4\\{ - 2}&{ - 7}&3&{ - 5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5\\{ - 1}\\3\\{ - 2}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 8}\\{16}\end{array}} \right]\)

Short Answer

Expert verified

The matrix equation as a vector equation is\(5 \cdot \left[ {\begin{array}{*{20}{c}}5\\{ - 2}\end{array}} \right] - 1 \cdot \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\end{array}} \right] + 3 \cdot \left[ {\begin{array}{*{20}{c}}{ - 8}\\3\end{array}} \right] - 2 \cdot \left[ {\begin{array}{*{20}{c}}4\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 8}\\{16}\end{array}} \right]\).

Step by step solution

01

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.

02

Write matrix A and vector x

Consider the equation \(\left[ {\begin{array}{*{20}{c}}5&1&{ - 8}&4\\{ - 2}&{ - 7}&3&{ - 5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5\\{ - 1}\\3\\{ - 2}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 8}\\{16}\end{array}} \right]\).

Here, \(A = \left[ {\begin{array}{*{20}{c}}5&1&{ - 8}&4\\{ - 2}&{ - 7}&3&{ - 5}\end{array}} \right]\), \({\bf{x}} = \left[ {\begin{array}{*{20}{c}}5\\{ - 1}\\3\\{ - 2}\end{array}} \right]\), and \(b = \left[ {\begin{array}{*{20}{c}}{ - 8}\\{16}\end{array}} \right]\).

03

Write matrix A columns and vector x entries

Here, \({{\bf{a}}_1} = \left[ {\begin{array}{*{20}{c}}5\\{ - 2}\end{array}} \right]\), \({{\bf{a}}_2} = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\end{array}} \right]\), \({{\bf{a}}_3} = \left[ {\begin{array}{*{20}{c}}{ - 8}\\3\end{array}} \right]\), \({{\bf{a}}_4} = \left[ {\begin{array}{*{20}{c}}4\\{ - 5}\end{array}} \right]\), \({x_1} = 5\), \({x_2} = - 1\), \({x_3} = 3\), and \({x_4} = - 2\).

04

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

Write the matrix equation \(\left[ {\begin{array}{*{20}{c}}5&1&{ - 8}&4\\{ - 2}&{ - 7}&3&{ - 5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5\\{ - 1}\\3\\{ - 2}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 8}\\{16}\end{array}} \right]\) as a vector equationby using the definition as shown below:

\(\begin{array}{c}\left[ {\begin{array}{*{20}{c}}5&1&{ - 8}&4\\{ - 2}&{ - 7}&3&{ - 5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5\\{ - 1}\\3\\{ - 2}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 8}\\{16}\end{array}} \right]\\5 \cdot \left[ {\begin{array}{*{20}{c}}5\\{ - 2}\end{array}} \right] - 1 \cdot \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\end{array}} \right] + 3 \cdot \left[ {\begin{array}{*{20}{c}}{ - 8}\\3\end{array}} \right] - 2 \cdot \left[ {\begin{array}{*{20}{c}}4\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 8}\\{16}\end{array}} \right]\end{array}\)

Thus, the matrix equation \(\left[ {\begin{array}{*{20}{c}}5&1&{ - 8}&4\\{ - 2}&{ - 7}&3&{ - 5}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5\\{ - 1}\\3\\{ - 2}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 8}\\{16}\end{array}} \right]\)can be written as a vector equation as \(5 \cdot \left[ {\begin{array}{*{20}{c}}5\\{ - 2}\end{array}} \right] - 1 \cdot \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\end{array}} \right] + 3 \cdot \left[ {\begin{array}{*{20}{c}}{ - 8}\\3\end{array}} \right] - 2 \cdot \left[ {\begin{array}{*{20}{c}}4\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - 8}\\{16}\end{array}} \right]\).

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Most popular questions from this chapter

Let a and b represent real numbers. Describe the possible solution sets of the (linear) equation \(ax = b\). (Hint:The number of solutions depends upon a and b.)

Consider the problem of determining whether the following system of equations is consistent for all \({b_1},{b_2},{b_3}\):

\(\begin{aligned}{c}{\bf{2}}{x_1} - {\bf{4}}{x_2} - {\bf{2}}{x_3} = {b_1}\\ - {\bf{5}}{x_1} + {x_2} + {x_3} = {b_2}\\{\bf{7}}{x_1} - {\bf{5}}{x_2} - {\bf{3}}{x_3} = {b_3}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of Span \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\). Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase 鈥渃olumns of A.鈥�
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

Consider a dynamical system x鈫�(t+1)=Ax鈫�(t) with two components. The accompanying sketch shows the initial state vector x鈫�0and two eigen vectors 蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫� of A (with eigen values 位1鈫�and位2鈫� respectively). For the given values of 位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=1,位2鈫�=0.9

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

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

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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