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Chapter 4: Q8Q (page 191)

In Exercises 5-8, find the coordinate vector \({\left[ {\mathop{\rm x}\nolimits} \right]_B}\) of \({\mathop{\rm x}\nolimits} \) relative to the given basis \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\).

8. \({{\mathop{\rm b}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\0\\3\end{array}} \right],{{\mathop{\rm b}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}2\\1\\8\end{array}} \right],{{\mathop{\rm b}\nolimits} _3} = \left[ {\begin{array}{*{20}{c}}1\\{ - 1}\\2\end{array}} \right],{\mathop{\rm x}\nolimits} = \left[ {\begin{array}{*{20}{c}}3\\{ - 5}\\4\end{array}} \right]\)

Short Answer

Expert verified

The coordinate vector \({\left[ {\mathop{\rm x}\nolimits} \right]_B}\) of x relative to the given basis is \({\left[ {\mathop{\rm x}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\5\end{array}} \right]\).

Step by step solution

01

Define the coordinate vector of x

Suppose \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\) is a basis for \(V\) and x is in \(V\). Thecoordinatesof \({\mathop{\rm x}\nolimits} \) relative to basis \(B\) (or the \(B\)-coordinates of x) are the weights \({c_1},...,{c_n}\), such that \({\mathop{\rm x}\nolimits} = {c_1}{b_1} + ... + {c_n}{b_n}\).

02

Write an augmented matrix

Consider the augmented matrix shown below.

\(\left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm b}\nolimits} _1}}&{{{\mathop{\rm b}\nolimits} _2}}&{{{\mathop{\rm b}\nolimits} _3}}&{\mathop{\rm x}\nolimits} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&2&1&3\\0&1&{ - 1}&{ - 5}\\3&8&2&4\end{array}} \right]\)

03

Apply the row operation

At row 3, multiply row 1 by 3 and subtract it from row 3. At row 1, multiply row 2 by 2 and subtract it from row 1.

\[ \sim \left[ {\begin{array}{*{20}{c}}1&0&3&{13}\\0&1&{ - 1}&{ - 5}\\0&2&{ - 1}&{ - 5}\end{array}} \right]\]

At row 3, multiply row 2 by 2 and subtract it from row 3.

\[ \sim \left[ {\begin{array}{*{20}{c}}1&0&3&{13}\\0&1&{ - 1}&{ - 5}\\0&0&1&5\end{array}} \right]\]

At row 1, multiply row 3 by 3 and subtract it from row 1. At row 2, multiply row 2 by 1 and add it to row 3.

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

04

Determine the coordinate vector \({\left[ {\mathop{\rm x}\nolimits}  \right]_B}\) of \({\mathop{\rm x}\nolimits} \)

The solution of matrix A is \({c_1} = - 2,{c_2} = 0,{c_3} = 5\).

The coordinates vector \({\left[ {\mathop{\rm x}\nolimits} \right]_B}\) of \({\mathop{\rm x}\nolimits} \) relative to the given basis is shown below:

\(\begin{array}{c}{\left[ {\mathop{\rm x}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}{{c_1}}\\{{c_2}}\\{{c_3}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\5\end{array}} \right]\end{array}\)

Thus, the coordinates vector \({\left[ {\mathop{\rm x}\nolimits} \right]_B}\) of \({\mathop{\rm x}\nolimits} \) relative to the given basis is \({\left[ {\mathop{\rm x}\nolimits} \right]_B} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\5\end{array}} \right]\).

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Most popular questions from this chapter

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

16. If \(A\) is an \(m \times n\) matrix of rank\(r\), then a rank factorization of \(A\) is an equation of the form \(A = CR\), where \(C\) is an \(m \times r\) matrix of rank\(r\) and \(R\) is an \(r \times n\) matrix of rank \(r\). Such a factorization always exists (Exercise 38 in Section 4.6). Given any two \(m \times n\) matrices \(A\) and \(B\), use rank factorizations of \(A\) and \(B\) to prove that rank\(\left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

(Hint: Write \(A + B\) as the product of two partitioned matrices.)

In Exercise 18, Ais an \(m \times n\) matrix. Mark each statement True or False. Justify each answer.

18. a. If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.

b. Row operations preserve the linear dependence relations among the rows of A.

c. The dimension of the null space of A is the number of columns of A that are not pivot columns.

d. The row space of \({A^T}\) is the same as the column space of A.

e. If A and B are row equivalent, then their row spaces are the same.

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

22. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 1}&{ - 13}&{ - 12.2}&{ - 1.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

Consider the polynomials \({{\bf{p}}_{\bf{1}}}\left( t \right) = {\bf{1}} + t\), \({{\bf{p}}_{\bf{2}}}\left( t \right) = {\bf{1}} - t\), \({{\bf{p}}_{\bf{3}}}\left( t \right) = {\bf{4}}\), \({{\bf{p}}_{\bf{4}}}\left( t \right) = {\bf{1}} + {t^{\bf{2}}}\), and \({{\bf{p}}_{\bf{5}}}\left( t \right) = {\bf{1}} + {\bf{2}}t + {t^{\bf{2}}}\), and let H be the subspace of \({P_{\bf{5}}}\) spanned by the set \(S = \left\{ {{{\bf{p}}_{\bf{1}}},\,{{\bf{p}}_{\bf{2}}},\;{{\bf{p}}_{\bf{3}}},\,{{\bf{p}}_{\bf{4}}},\,{{\bf{p}}_{\bf{5}}}} \right\}\). Use the method described in the proof of the Spanning Set Theorem (Section 4.3) to produce a basis for H. (Explain how to select appropriate members of S.)

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

13. Show that if \(P\) is an invertible \(m \times m\) matrix, then rank\(PA\)=rank\(A\).(Hint: Apply Exercise12 to \(PA\) and \({P^{ - 1}}\left( {PA} \right)\).)
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