Question

# 4. Repeated Application of a Linear Map: Consider the linear map T: RS → RS represented in matrix form on the standard basis as M[T]=\left[\begin{array}{rrrrc} 1 & 0 & -1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ -1 & 0 & -1 & -1 & -1 \\ 1 & 0 & -1 & 0 & 1 \\ 1 & 1 & -1 & 1 & -1 \end{array}\right] This problem will investigate what happens when a linear map (in this case T) is repeatedly applied to a vector.Important Note: The map T, like most maps, does not maintain the length of a vector.Repeated application can result in very large (or very small) values for some of the coordinates.To avoid complications introduced by this effect, we will be frequently normalizing the vectors as we proceed. For the purposes of this exercise, normalizing a vector is defined as dividing the vector by the vector element with the largest absolute value. That is, the vector is scaled by an amount that turns the element with the largest absolute value into the value 1. a) Write a MATLAB script that does the following:• Generate a random vector, v, in R³ and normalize it. Feel free to constrain your random number selection to the default domain of the rand command.•Apply the map T to the normalized input vector. Normalize and store the result.Now apply the map T to this normalized output vector. Normalize and store the result. Repeat in this manner until you have generated the output for 1-25 repeated applications of T to a given random initial vector. b) Make a plot that compares the results for 1, 3, 5, 10, and 25 applications of T to a given random initial vector. Comment on the results. c) Make a plot that compares the results from 25 applications of T to 5 different random initial vectors. Comment on the results. d) In class we are just starting to learn how to find eigenvectors and eigenvalues of a linear map. However, for this exercise, we will allow you to use the numerical routine in MATLAB. The form of the command is [V,D] = eig(T). The routine will return two matrices. V is a matrix whose columns are the eigenvectors of T, and D is a diagonal matrix whose values are the corresponding eigenvalues, i.e., the eigenvalue in column n is associated with the eigenvector in column n. Modify your script to find the eigenvalues and eigenvectors of T. Locate the eigenvalue with the largest absolute value and call its corresponding eigenvector w. Normalize w. e) Make a plot that compares the normalized version of w to your results for 1, 5, and 10 applications of T to a given random initial vector. Comment on the results and argue intuitively about how they are related to the concept of an invariant subspace.  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8