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Sums and Products of Linear Transformations

  • Thomas Banchoff
  • John Wermer
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

If T and S are linear transformations, then we may define a new transformation T + S by the condition
$$ (T\; + \;S)(X)\;{\rm{ = }}\;T(X)\;{\rm{ + }}\;S(X)\;\;\;\;\;{\rm{for every vector }}X. $$
Then by definition, (T + S)(X + Y) = T(X + Y) + S(X + Y), and since T and S are linear transformations, this equals T(X) + T(Y) + S(X) + S(Y) = T(X) + S(X) + T(Y) + S(Y) = (T + S)(X) + (T + S)(Y). Thus for every pair X, Y, we have
$$ (T\;{\rm{ + }}\;S)(X\;{\rm{ + }}\;Y) = (T\;{\rm{ + }}\;S)(X) + (T\;{\rm{ + }}\;S)(Y). $$

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Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Thomas Banchoff
    • 1
  • John Wermer
    • 1
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

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