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Chapter 5: Q35E (page 267)

Question: Let \({\rm{u}}\) and \({\rm{v}}\) be the vectors shown in the figure, and suppose \({\rm{u}}\) and \({\rm{v}}\) are eigenvectors of a \(2 \times 2\) matrix A that correspond to eigenvalues 2 and 3, respectively. Let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be the linear transformation given by \(T\left( x \right) = Ax\) for each \({\rm{x}}\) in \({\mathbb{R}^2}\), and let \({\rm{w}} = {\rm{u}} + {\rm{v}}\). Make a copy of the figure, and on the same coordinate system, carefully plot the vectors \(T\left( {\rm{u}} \right)\), \(T\left( {\rm{v}} \right)\), and \(T\left( {\rm{w}} \right)\).

Short Answer

Expert verified

The image is given below:

Step by step solution

01

Plot \(T\left( {\rm{u}} \right) = {\rm{2u}}\)

With the help of the given figure in the exercise, draw \(2u\) because it is given that \(u\) is an eigenvector of matrix A that corresponds to eigenvalue 2.

02

Plot \(T\left( {\rm{v}} \right) = 3{\rm{v}}\)

Similarly, with the help of the given figure in the exercise, draw \(3v\) because it is given that \(u\) is an eigenvector of matrix A that corresponds to eigenvalue 3.

03

Plot \(T\left( {\rm{w}} \right)\)

It is given that \(w = u + v\), since \(T\) is linear transformation given by \(T\left( x \right) = Ax\).

So, \(T\left( w \right)\) can be obtained as follows:

\(\begin{array}{c}T\left( w \right) = T\left( {u + v} \right)\\ = A\left( {u + v} \right)\\ = A\left( u \right) + A\left( v \right)\\ = T\left( u \right) + T\left( v \right)\end{array}\)

Now, plot \(T\left( w \right) = T\left( u \right) + T\left( v \right)\) as given in the image below:

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Most popular questions from this chapter

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: Let \(A = \left( {\begin{array}{*{20}{c}}{.5}&{.2}&{.3}\\{.3}&{.8}&{.3}\\{.2}&0&{.4}\end{array}} \right)\), \({{\rm{v}}_1} = \left( {\begin{array}{*{20}{c}}{.3}\\{.6}\\{.1}\end{array}} \right)\), \({{\rm{v}}_2} = \left( {\begin{array}{*{20}{c}}1\\{ - 3}\\2\end{array}} \right)\), \({{\rm{v}}_3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\0\\1\end{array}} \right)\) and \({\rm{w}} = \left( {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right)\).

  1. Show that \({{\rm{v}}_1}\), \({{\rm{v}}_2}\), and \({{\rm{v}}_3}\) are eigenvectors of \(A\). (Note: \(A\) is the stochastic matrix studied in Example 3 of Section 4.9.)
  2. Let \({{\rm{x}}_0}\) be any vector in \({\mathbb{R}^3}\) with non-negative entries whose sum is 1. (In section 4.9, \({{\rm{x}}_0}\) was called a probability vector.) Explain why there are constants \({c_1}\), \({c_2}\), and \({c_3}\) such that \({{\rm{x}}_0} = {c_1}{{\rm{v}}_1} + {c_2}{{\rm{v}}_2} + {c_3}{{\rm{v}}_3}\). Compute \({{\rm{w}}^T}{{\rm{x}}_0}\), and deduce that \({c_1} = 1\).
  3. For \(k = 1,2, \ldots ,\) define \({{\rm{x}}_k} = {A^k}{{\rm{x}}_0}\), with \({{\rm{x}}_0}\) as in part (b). Show that \({{\rm{x}}_k} \to {{\rm{v}}_1}\) as \(k\) increases.

Consider the growth of a lilac bush. The state of this lilac bush for several years (at year鈥檚 end) is shown in the accompanying sketch. Let n(t) be the number of new branches (grown in the year t) and a(t) the number of old branches. In the sketch, the new branches are represented by shorter lines. Each old branch will grow two new branches in the following year. We assume that no branches ever die.

(a) Find the matrix A such that [nt+1at+1]=A[ntat]

(b) Verify that [11]and [2-1] are eigenvectors of A. Find the associated eigenvalues.

(c) Find closed formulas for n(t) and a(t).

Question: Is \(\lambda = - 2\) an eigenvalue of \(\left( {\begin{array}{*{20}{c}}7&3\\3&{ - 1}\end{array}} \right)\)? Why or why not?

Let\(B = \left\{ {{{\bf{b}}_1},{{\bf{b}}_2},{{\bf{b}}_3}} \right\}\) be a basis for a vector space\(V\). Find \(T\left( {3{{\bf{b}}_1} - 4{{\bf{b}}_2}} \right)\) when \(T\) isa linear transformation from \(V\) to \(V\) whose matrix relative to \(B\) is

\({\left( T \right)_B} = \left( {\begin{aligned}0&{}&{ - 6}&{}&1\\0&{}&5&{}&{ - 1}\\1&{}&{ - 2}&{}&7\end{aligned}} \right)\)
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