Linear mappings from one vector space to another have acquired an important role in mathematics. Each linear transformation can be represented by a matrix, which enables us to work with the matrix in place of the transformation.

Definition 12.5 Let U and V be vector spaces over the same field F. A mapping T: UV is said to be a linear transformation if

  1. T(xb + yb) = Txb + T,    ∀ , ∈ U
  2. T(α) = αT,     ∀ αF, ∈ U

The conditions ...

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