Linear Algebra and the C Language/a0ms

Matrix of a linear application with respect to the basis B:

  Let B be the change of basis matrix of the basis B:   
  
  We know that:
  
    1) x_b = InvB x_s           
    2) x_s =    B x_b

  Let     T(x_s) = A x_s be a linear application with respect to the canonical basis. (standard)

  Let [T(x_b)]_b = D x_b the linear application  with respect to the basis B.

  Study the relationship between the matrix A and the matrix D.
 
  We have:
  
   T(x_s)    =  A x_s                                                  0) In standard basis
  [T(x_s)]_b = [A x_s]_b = InvB (A   x_s)                              1) x_b = InvB x_s 
  [T(x_s)]_b =            (InvB  A)  x_s                     
  [T(x_b)]_b =            (InvB  A) (B x_b)                            2) x_s =    B x_b

  [T(x_b)]_b = (InvB A)(B  x_b) 
  [T(x_b)]_b = (InvB A  B) x_b   

  [T(x_b]_b =      D       x_b             The linear application [T(x_b)]_b with respect to the basis B.
  
         D = (InvB A  B)                   This is the relationship between the matrix A and the matrix D

...a.c : Calculate vectors x_b and x_s
...b.c : Calculate T(x) = A x_b
...c.c : Calculate [T(x_b)]_b = D x_b
...d.c : Check if: B T(x_b)_b = T(x_s)

Example in R2:


* c02a.c	* c02b.c	* c02c.c	* c02d.c

Example in R3:


* c03a.c	* c03b.c	* c03c.c	* c03d.c

Example in R4:


* c04a.c	* c04b.c	* c04c.c	* c04d.c