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Chapter 1: Q26E (page 1)

Let \(A = \left[ {\begin{array}{*{20}{c}}2&0&6\\{ - 1}&8&5\\1&{ - 2}&1\end{array}} \right]\), let \(b = \left[ {\begin{array}{*{20}{c}}{10}\\3\\3\end{array}} \right]\) , and let \(W\) be the set of all linear combinations of the columns of \(A\).

  1. Is \(b\) in \(W\)?
  2. Show that the third column of \(A\) is in \(W\).

Short Answer

Expert verified
  1. \(b\)is a linear combination of \(A\), which means \(b\) is in \(W\).
  2. The third column \({a_3}\) is in \(W\).

Step by step solution

01

Write the matrix into an augmented matrix

Avector equation \({x_1}{a_1} + {x_2}{a_2} + ... + {x_n}{a_n} = b\) has the same solution set as the linear system whose augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{....}&{{a_n}}&b\end{array}} \right]\). In particular, \(b\) can be generated by a linear combination of \({{\mathop{\rm a}\nolimits} _1},...{a_n}\), if and only if, there exists a solution to the linear system corresponding to the augmented matrix.

Write the given matrix into an augmented matrix.

\(\left[ {\begin{array}{*{20}{c}}2&0&6&{10}\\{ - 1}&8&5&3\\1&{ - 2}&1&3\end{array}} \right]\)

02

Apply row operation to identify that \(b\) is in \(W\)

a.

Perform an elementary row operation to produce the first augmented matrix.

Multiply row 1 by \(\frac{1}{2}\).

\(\left[ {\begin{array}{*{20}{c}}1&0&3&5\\{ - 1}&8&5&3\\1&{ - 2}&1&3\end{array}} \right]\)

Perform an elementary row operation to produce the second augmented matrix.

Add 1 time row 1 to row 2 and add \( - 1\) times row 1 to row 3.

\(\left[ {\begin{array}{*{20}{c}}1&0&3&5\\0&8&8&8\\0&{ - 2}&{ - 2}&{ - 2}\end{array}} \right]\)

03

Apply row operation to identify that \(b\) is in \(W\)

Perform an elementaryrow operation to produce the third augmented matrix.

Multiply row 2 by \(\frac{1}{8}\).

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

Apply sum of 2 times of row 2 and row 3 at row 3

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

When \({a_1},{a_2}\), and \({a_3}\) are the three columns of \(A\), the system of equations corresponding to the vector equation \({x_1}{a_1} + {x_2}{a_2} + {x_3}{a_3} = b\) is consistent.

Thus, \(b\) is a linear combination of \(A\), which means \(b\) is in \(W\).

04

Show that the third column of A is in W

b.

If vector\({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\)in\({\mathbb{R}^n}\)are given with scalars\({c_1},{c_2},...,{c_p}\), then the vector\({\mathop{\rm y}\nolimits} \)defined by\(y = {c_1}{v_1} + .... + {c_p}{v_p}\)is called alinear combination of\({{\mathop{\rm v}\nolimits} _1},{v_2},...,{v_p}\)with weights\({c_1},{c_2},...,{c_p}\).

Let\({a_1},{a_2}\), and \({a_3}\) represent the three columns of \(A\). Since \({{\mathop{\rm a}\nolimits} _3} = 0{a_1} + 0{a_2} + 1{a_3}\) is a linear combination of the columns of \(A\), so \({a_3}\) is in \(W\).

Hence, the third column \({a_3}\) is in \(W\).

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Most popular questions from this chapter

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

14. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 6}\\0&3&7\\1&{ - 2}&5\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}{11}\\{ - 5}\\9\end{array}} \right]\)

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. (Hint: Given u, v in \({\mathbb{R}^n}\), let \({\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\). Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \(T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \). Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).)

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 eigenvectors 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�=1.2,位2鈫�=1.1

Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.

17. \(\left[ {\begin{array}{*{20}{c}}2&3&h\\4&6&7\end{array}} \right]\)

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation, and let Sand U be functions from \({\mathbb{R}^n}\) into \({\mathbb{R}^n}\) such that \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) and \(\)\(U\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\). Show that \(U\left( v \right) = S\left( v \right)\) for all v in \({\mathbb{R}^n}\). This will show that Thas a unique inverse, as asserted in theorem 9. (Hint: Given any v in \({\mathbb{R}^n}\), we can write \({\mathop{\rm v}\nolimits} = T\left( {\mathop{\rm x}\nolimits} \right)\) for some x. Why? Compute \(S\left( {\mathop{\rm v}\nolimits} \right)\) and \(U\left( {\mathop{\rm v}\nolimits} \right)\)).
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