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Chapter 5: Q5.4-27E (page 267)

Let \(V\) be \({\mathbb{R}^n}\) with a basis \(B = \left\{ {{b_1},........{b_n}} \right\}\); let \(W\) be \({\mathbb{R}^n}\) with the standard basis, denoted here by \(\xi \); and consider the identity transformation \(I:V \to W\) , where \(I\left( {\rm{x}} \right) = {\rm{x}}\). Find the matrix for \(I\) relative to \(B\) and \(\xi \). What was this matrix called in section 4.4?

Short Answer

Expert verified

The matrix \(I\) relative to \(B\)and \(\xi \) is the set of vectors \(\left( {{{\bf{b}}_1}\,\,\,{{\bf{b}}_2}........\,{{\bf{b}}_n}} \right)\). The matrix \(\left( {{{\bf{b}}_1}\,\,\,{{\bf{b}}_2}........\,{{\bf{b}}_n}} \right)\) is called the change of coordinates matrix.

Step by step solution

01

Use the given information 

It is given that \(I:V \to W\), such that, \(I\left( {\bf{x}} \right) = {\bf{x}}\) . So, for each \({j^{th}}\) vector of the basis \(B\) and \(\xi \), the identity \(I\left( {\bf{x}} \right) = {\bf{x}}\) must be satisfied, such that \({\left( {I\left( {{{\bf{b}}_j}} \right)} \right)_\xi } = {{\bf{b}}_j}\). This implies that matrix \(I\) relative to \(B\)and \(\xi \) is the set of vectors \(\left( {{{\bf{b}}_1}\,\,\,{{\bf{b}}_2}........\,{{\bf{b}}_n}} \right)\).

02

Define the matrix

In section 4.4, the change of coordinates matrix is defined, which changes the \(B\)-coordinates of a vector \(x\) into the standard coordinates for \(x\), where the change is carried out in \({\mathbb{R}^n}\)for the basis \(B = \left( {{{\bf{b}}_1}\,\,\,{{\bf{b}}_2}........\,{{\bf{b}}_n}} \right)\). So, \(\left( {{{\bf{b}}_1}\,\,\,{{\bf{b}}_2}........\,{{\bf{b}}_n}} \right)\) is called the change of coordinates matrix.

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Most popular questions from this chapter

a. Let \(A\) be a diagonalizable \(n \times n\) matrix. Show that if the multiplicity of an eigenvalue \(\lambda \) is \(n\), then \(A = \lambda I\).

b. Use part (a) to show that the matrix \(A =\left({\begin{aligned}{*{20}{l}}3&1\\0&3\end{aligned}}\right)\) is not diagonalizable.

Suppose \(A\) is diagonalizable and \(p\left( t \right)\) is the characteristic polynomial of \(A\). Define \(p\left( A \right)\) as in Exercise 5, and show that \(p\left( A \right)\) is the zero matrix. This fact, which is also true for any square matrix, is called the Cayley-Hamilton theorem.

Question: Diagonalize the matrices in Exercises \({\bf{7--20}}\), if possible. The eigenvalues for Exercises \({\bf{11--16}}\) are as follows:\(\left( {{\bf{11}}} \right)\lambda {\bf{ = 1,2,3}}\); \(\left( {{\bf{12}}} \right)\lambda {\bf{ = 2,8}}\); \(\left( {{\bf{13}}} \right)\lambda {\bf{ = 5,1}}\); \(\left( {{\bf{14}}} \right)\lambda {\bf{ = 5,4}}\); \(\left( {{\bf{15}}} \right)\lambda {\bf{ = 3,1}}\); \(\left( {{\bf{16}}} \right)\lambda {\bf{ = 2,1}}\). For exercise \({\bf{18}}\), one eigenvalue is \(\lambda {\bf{ = 5}}\) and one eigenvector is \(\left( {{\bf{ - 2,}}\;{\bf{1,}}\;{\bf{2}}} \right)\).

12. \(\left( {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{2}}&{\bf{2}}\\{\bf{2}}&{\bf{4}}&{\bf{2}}\\{\bf{2}}&{\bf{2}}&{\bf{4}}\end{array}} \right)\)

Question: Exercises 9-14 require techniques section 3.1. Find the characteristic polynomial of each matrix, using either a cofactor expansion or the special formula for \(3 \times 3\) determinants described prior to Exercise 15-18 in Section 3.1. [Note: Finding the characteristic polynomial of a \(3 \times 3\) matrix is not easy to do with just row operations, because the variable \(\lambda \) is involved.

12. \(\left[ {\begin{array}{*{20}{c}}- 1&0&1\\- 3&4&1\\0&0&2\end{array}} \right]\)

(M)Use a matrix program to diagonalize

\(A = \left( {\begin{aligned}{*{20}{c}}{ - 3}&{ - 2}&0\\{14}&7&{ - 1}\\{ - 6}&{ - 3}&1\end{aligned}} \right)\)

If possible. Use the eigenvalue command to create the diagonal matrix \(D\). If the program has a command that produces eigenvectors, use it to create an invertible matrix \(P\). Then compute \(AP - PD\) and \(PD{P^{{\bf{ - 1}}}}\). Discuss your results.
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