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For the linear transformation given by
find the matrix for with respect to the bases

Ask by Mitchell Sherman. in Russia
Dec 16,2024

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The matrix for the linear transformation is:

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To find the matrix for the linear transformation with respect to the bases and , we follow these steps:
  1. Write the Transformation: The linear transformation is given by
  2. Compute the Images of the Basis Vectors: We will apply to each vector in the basis .
    • For :
    • For :
  3. Express Images in Terms of the Basis : Next, we express the images in terms of the basis .
    • For :
      We set up the equations:
      1. (since the third component is 0)
      From the third equation, . Substituting into the second equation gives:
      Substituting into the first equation:
      Therefore, in terms of the basis :
    • For :
      Setting up the equations:
      From the third equation, . Using this in the second equation gives:
      Using this in the first equation:
      Therefore, in terms of the basis :
  4. Construct the Matrix: The resulting vectors from the transformation can be arranged as columns in the matrix:
    [
    A_{

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