This question is on eigenvalues eigenvectors and diagonaliza

This question is on eigenvalues, eigenvectors and diagonalization: David Cherney and Tom Denton\'s \"Linear Algebra\" Chapter# 13

2. When writing a matrix for a linear transformation, we have seen that the choice of basis matters. In fact, even the order of the basis matters! the choice of basis matters. In fact, even the order of the basis matte (a) Write all possible reorderings of the standard basis (e1,e2, es) for R3. b) Write each change of basis matrix between the standard basis and each of its reorderings. Make as many observations as you can about these matrices. what are their entries? Do you notice anything about how many of each type of entry appears in each row and column? What are their determinants? (Note: These matrices are known as permutation matrices.) (c) Given L : R3 R3 is linear and 2a 3r 2z+x+y write the matrix M for L in the standard basis, and two reorder- ings of the standard basis. How are these matrices related?

Solution

a) There are possible 3 places and 3 vectors e1,e2 and e3

Hence the first position can be filled in 3 ways

Hence the second position can be filled in 2 ways

Hence the last position can be filled in 1 ways

Hence number of re-ordering possible = 3 * 2 * 1 = 3! = 6 ways

The possible re-orderings are (e1,e2,e3), (e1,e3,e2), (e2,e1,e3), (e2,e3,e1), (e3,e1,e2), (e3,e2,e1)

b)

The first matrix will be [1 0 0; 0 1 0; 0 0 1]

The second matrix will be [1 0 0; 0 0 1; 0 1 0]

The third matrix will be [0 1 0; 1 0 0; 0 0 1]

The fourth matrix will be [0 1 0; 0 0 1; 1 0 0]

The fifth matrix will be [0 0 1;1 0 0;0 1 0]

The sixth matrix will be [0 0 1; 0 1 0; 1 0 0]

c)

The matrix M will be a 3X3 matrix

ax + by + cz = (2y-z), a=0,b=2,c=-1

dx + ey + fz = 3x, d=3,e=0,f=0

gx + hy + iz = 2z + x + y, g=1,h=1,i=2

Hence the matrix will be

a	b	c
d	e	f
g	h	i