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Chapter 4: Q9Q (page 191)

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

9. \(B = \left\{ {\left[ {\begin{array}{*{20}{c}}2\\{ - 9}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}1\\8\end{array}} \right]} \right\}\)

Short Answer

Expert verified

The change-of-coordinates matrix from \(B\) to the standard basis in \({\mathbb{R}^2}\) is \({P_B} = \left[ {\begin{array}{*{20}{c}}2&1\\{ - 9}&8\end{array}} \right]\).

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 \({\mathbb{R}^2}\) is shown below:

\(\begin{array}{c}{P_B} = \left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm b}\nolimits} _1}}&{{{\mathop{\rm b}\nolimits} _2}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}2&1\\{ - 9}&8\end{array}} \right]\end{array}\)

Thus, the change-of-coordinates matrix from \(B\) to the standard basis in \({\mathbb{R}^2}\) is \({P_B} = \left[ {\begin{array}{*{20}{c}}2&1\\{ - 9}&8\end{array}} \right]\).

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Most popular questions from this chapter

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.

a. What is the dimension of range of T if T is one-to-one mapping? Explain.

b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.

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 Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work

\({\left( {{\bf{1}} - t} \right)^{\bf{2}}}\),\(t - {\bf{2}}{t^{\bf{2}}} + {t^{\bf{3}}}\),\({\left( {{\bf{1}} - t} \right)^{\bf{3}}}\)

[M] Let \(A = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&{ - 4}&5&3&{ - 3}&{ - 7}\\{ - 4}&6&7&{ - 2}&{ - 6}&{ - 5}&5\\5&{ - 7}&{ - 6}&5&{ - 6}&2&8\\{ - 3}&5&8&{ - 1}&{ - 7}&{ - 4}&8\\6&{ - 8}&{ - 5}&4&4&9&3\end{array}} \right]\).

  1. Construct matrices \(C\) and \(N\) whose columns are bases for \({\mathop{\rm Col}\nolimits} A\) and \({\mathop{\rm Nul}\nolimits} A\), respectively, and construct a matrix \(R\) whose rows form a basis for Row\(A\).
  2. Construct a matrix \(M\) whose columns form a basis for \({\mathop{\rm Nul}\nolimits} {A^T}\), form the matrices \(S = \left[ {\begin{array}{*{20}{c}}{{R^T}}&N\end{array}} \right]\) and \(T = \left[ {\begin{array}{*{20}{c}}C&M\end{array}} \right]\), and explain why \(S\) and \(T\) should be square. Verify that both \(S\) and \(T\) are invertible.

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

21. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 2}&{ - 4.2}&{ - 4.8}&{ - 3.6}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).
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