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Chapter 4: Q10Q (page 191)

In Exercises 9 and 10, find the change-of-coordinates matrix from \(B\) to the standard basis in \({\mathbb{R}^n}\).

10. \(B = \left\{ {\left[ {\begin{array}{*{20}{c}}3\\{ - 1}\\4\end{array}} \right],\left[ {\begin{array}{*{20}{c}}2\\0\\{ - 5}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}8\\{ - 2}\\7\end{array}} \right]} \right\}\)

Short Answer

Expert verified

The change-of-coordinates matrix from \(B\) to the standard basis in \({\mathbb{R}^3}\) is \({P_B} = \left[ {\begin{array}{*{20}{c}}3&2&8\\{ - 1}&0&{ - 2}\\4&{ - 5}&7\end{array}} \right]\).

Step by step solution

01

State the change of coordinates

Let\({P_B} = \left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm b}\nolimits} _1}}&{{{\mathop{\rm b}\nolimits} _2}}& \cdots &{{{\mathop{\rm b}\nolimits} _n}}\end{array}} \right]\), then thevector equation\[{\mathop{\rm x}\nolimits} = {c_1}{{\mathop{\rm b}\nolimits} _1} + {c_2}{{\mathop{\rm b}\nolimits} _2} + ... + {c_n}{{\mathop{\rm b}\nolimits} _n}\] becomes equivalent to \({\mathop{\rm x}\nolimits} = {P_B}{\left[ {\mathop{\rm x}\nolimits} \right]_B}\). \({P_B}\) denotes thechange-of-coordinates matrixfrom \(B\) to the standard basis in \({\mathbb{R}^n}\).

02

Determine the change-of-coordinates matrix from \(B\)

The change-of-coordinates matrix from\(B\)to the standard basis in\({\mathbb{R}^3}\)is shown below.

\[\begin{array}{c}{P_B} = \left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm b}\nolimits} _1}}&{{{\mathop{\rm b}\nolimits} _2}}&{{{\mathop{\rm b}\nolimits} _3}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}3&2&8\\{ - 1}&0&{ - 2}\\4&{ - 5}&7\end{array}} \right]\end{array}\]

Thus, the change-of-coordinates matrix from \(B\) to the standard basis in \({\mathbb{R}^3}\) is \({P_B} = \left[ {\begin{array}{*{20}{c}}3&2&8\\{ - 1}&0&{ - 2}\\4&{ - 5}&7\end{array}} \right]\).

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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 \(m \times n\).

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

Let \(A\) be an \(m \times n\) matrix of rank \(r > 0\) and let \(U\) be an echelon form of \(A\). Explain why there exists an invertible matrix \(E\) such that \(A = EU\), and use this factorization to write \(A\) as the sum of \(r\) rank 1 matrices. [Hint: See Theorem 10 in Section 2.4.]

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

  1. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} A\). (Hint: Explain why every vector in the column space of \(AB\) is in the column space of \(A\).
  2. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} B\). (Hint: Use part (a) to study rank\({\left( {AB} \right)^T}\).)

In Exercise 2, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

2. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{5}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{7}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{8}}\\{ - {\bf{5}}}\end{array}} \right)\)

Suppose \(A\) is \(m \times n\)and \(b\) is in \({\mathbb{R}^m}\). What has to be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent?
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