Matrix of a linear application

Linear Algebra and the C Language · Wikibooks

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

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

  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                 In standard basis
                                                                         
  [T(x_s)]_b = [A x_s]_b              In the B basis

  We know that: [U]_b = InvB [U]_s then [A x_s]_b = InvB [A x_s]_s = InvB A x_s

  [T(x_s)]_b = InvB (A x_s)        

  [T(x_s)]_b = (InvB A)(x_s)      

  We know that: U_s = B U_b then (x_s) = (B x_b)
              
  [T(x_b)]_b = (InvB A)(B x_b)        
  [T(x_b)]_b = (InvB A B) x_b 
 
  D = (InvB A B) is the relationship between A and 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