Minh Nguyen							mtn8286
Week 11								7.2.8

http://calclab.math.tamu.edu/~mtn8286/7_2_8.html

Problem:

Determinants are defined for matrices, but at some points in this section
we have referred to the determinant of a linear function, thereby tacitly 
assuming that the determinant does not depend on which matrix representation
of the function is used to calculate the determinant.

  (a) Show that this assumption is justified for a linear function 
      L : V ®V, with the understanding that the basis used for V in its 
      role as domain is the same as the basis used for V in its role as
      codomain.

  (b) Show why the assumption is not justified for a linear function 
      L : V ®W, where V and W are two unrelated vector spaces that simply
      happen to have the same dimension (so that the determinants of the 
      matrix representations of L are defined).
				

Solution:

Recall the change of basis theorem from Section 4.4 and 4.5.

Let V be an n-dimensional space for both domain and codomain.  Choose 
 as the basis for V (domain) and  as the basis 
for V (codomain). Let L be a linear function from V into V whose matrix 
with respect to the these bases is A.

	    

Choose the same new basis  for V(domain and codomain) 
related to the natural bases by a matrix G. 

		      

Then the matrix of L with respect to the new basis is:   G-1AG


If the determinant does not depend on the matrix representation of L, then	
		
		det A  =  det (G-1AG)

By the identity :  det (AB) = (det A)(det B)

		det A  =  det (G-1AG)  =  1/(det G)(det A)(det G)  =  det A

We will demonstrate det A  =  det (G-1AG) by computing the determinants of 
the following change-of-basis :








(b)

Let V be an n-dimensional space for the domain and W be an n-dimensional 
space for the codomain.  Choose a basis  for V and basis 
 for W.  (Let L be a linear function from V into W whose 
matrix with respect to the these bases is A. 

	    

Let  be a new basis for V related to the natural basis 
by a matrix G: 
		      ,

And  be a new basis for W related to the natural basis 
by a matrix H: 
			.

Then the matrix of L with respect to the new bases is:   H-1AG

If the determinant does depend on the matrix representation of L, then	
		
		det A  ¹  det (H-1AG)

By the identity :  det (AB) = (det A)(det B)

		det A  ¹  det (H-1AG) = 1/(det H)(det A)(det G) 

Again using the change-of-basis in (a) and introducing a new basis H ,
we will show det A  ¹  det (H-1AG):