Sums and Products of Linear Transformations

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

$$ (T\;{\rm{ + }}\;S)(X\;{\rm{ + }}\;Y) = (T\;{\rm{ + }}\;S)(X) + (T\;{\rm{ + }}\;S)(Y). $$