text: ### Change of Basis
  
  ##### 4. Change of the matrix for linear transformation
  
  The matrix [T(B)]C of a linear transformation T: VW depends on the choices of bases B and C. If B' and C' are other choices, then everything fits into a diagram (see this formula).
  
  From the red arrows in the diagram, we conclude the following (the second equality follows from this formula).
  
  For a linear transformation T, the matrices with regard to different bases are related by
  
  [T(B')]C' = [C]C' [T(B)]C [B]B'-1 = [C]C' [T(B)]C [B']B.
  
  Example For the linear transformation and bases in this example, we found the matrix for the transformation.
  
   [T(Hermite)]C = [ 1 0 -1 0 ] 1 1 0 -2 0 1 0 -3 0 1 0 -3
  
  Now we change the basis B to Chebyshev = {1, t, 2t2, 4t3 - 3t} and keep the basis C as before. To find the new matrix [T(Chebyshev)]C, we recall the matrix for the change of coordinates from another example.
  
   [Chebychev]Hermite = [ 1 0 2 0 ] 0 1 0 9 0 0 2 0 0 0 0 4
  
  Thus we have
  
   [T(Chebychev)]C = [T(Hermite)]C [Chebychev]Hermite = [ 1 0 0 0 ]. 1 1 2 1 0 1 0 -3 0 1 0 -3
  
  We may compare the result with the direct application of T to Chebychev.
  
   T(1) = [ 1 1 ], T(t) = [ 0 1 ], T(2t2) = [ 0 2 ], T(4t3 - 3t) = [ 0 1 ] 0 0 1 1 0 0 -3 -3
explanation: The document discusses the change of basis for a linear transformation matrix. It provides an example where the matrix of a linear transformation is computed with respect to different bases, specifically from the Hermite basis to the Chebyshev basis. The document includes the necessary matrices and the steps to compute the new matrix. It also provides the result of applying the transformation directly to the Chebyshev basis for verification.
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distilabel_metadata.content: ### Change of Basis
  
  ##### 4. Change of the matrix for linear transformation
  
  The matrix [T(B)]C of a linear transformation T: VW depends on the choices of bases B and C. If B' and C' are other choices, then everything fits into a diagram (see this formula).
  
  From the red arrows in the diagram, we conclude the following (the second equality follows from this formula).
  
  For a linear transformation T, the matrices with regard to different bases are related by
  
  [T(B')]C' = [C]C' [T(B)]C [B]B'-1 = [C]C' [T(B)]C [B']B.
  
  Example For the linear transformation and bases in this example, we found the matrix for the transformation.
  
   [T(Hermite)]C = [ 1 0 -1 0 ] 1 1 0 -2 0 1 0 -3 0 1 0 -3
  
  Now we change the basis B to Chebyshev = {1, t, 2t2, 4t3 - 3t} and keep the basis C as before. To find the new matrix [T(Chebyshev)]C, we recall the matrix for the change of coordinates from another example.
  
   [Chebychev]Hermite = [ 1 0 2 0 ] 0 1 0 9 0 0 2 0 0 0 0 4
  
  Thus we have
  
   [T(Chebychev)]C = [T(Hermite)]C [Chebychev]Hermite = [ 1 0 0 0 ]. 1 1 2 1 0 1 0 -3 0 1 0 -3
  
  We may compare the result with the direct application of T to Chebychev.
  
   T(1) = [ 1 1 ], T(t) = [ 0 1 ], T(2t2) = [ 0 2 ], T(4t3 - 3t) = [ 0 1 ] 0 0 1 1 0 0 -3 -3
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document discusses the change of basis for a linear transformation matrix. It provides an example where the matrix of a linear transformation is computed with respect to different bases, specifically from the Hermite basis to the Chebyshev basis. The document includes the necessary matrices and the steps to compute the new matrix. It also provides the result of applying the transformation directly to the Chebyshev basis for verification.
  
  Label: no-question