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The sum of linear transformations is a linear transformation

Created over 8 years ago, updated 10 days ago

Theorem: If (S, T: V o W) are linear transformations, then (S + T) is also a linear transformation.

Proof:

  1. Additivity: ((S+T)(\mathbf{u} + \mathbf{v}) = S(\mathbf{u} + \mathbf{v}) + T(\mathbf{u} + \mathbf{v}) = S(\mathbf{u}) + S(\mathbf{v}) + T(\mathbf{u}) + T(\mathbf{v}) = (S(\mathbf{u}) + T(\mathbf{u})) + (S(\mathbf{v}) + T(\mathbf{v})) = (S+T)(\mathbf{u}) + (S+T)(\mathbf{v}))

  2. Homogeneity: ((S+T)(c\mathbf{v}) = S(c\mathbf{v}) + T(c\mathbf{v}) = cS(\mathbf{v}) + cT(\mathbf{v}) = c(S(\mathbf{v}) + T(\mathbf{v})) = c(S+T)(\mathbf{v}))

Both linearity conditions are satisfied, so (S + T) is linear.

Intuition: The pointwise sum of two structure-preserving maps is itself structure-preserving because addition in the codomain is compatible with the linearity of each map.