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A scalar multiple of a linear transformation is a linear transformation

Created over 8 years ago, updated 10 days ago

Theorem: If (T: V o W) is a linear transformation and (c) is a scalar, then (cT) is also a linear transformation.

Proof:

  1. Additivity: ((cT)(\mathbf{u} + \mathbf{v}) = c \cdot T(\mathbf{u} + \mathbf{v}) = c(T(\mathbf{u}) + T(\mathbf{v})) = cT(\mathbf{u}) + cT(\mathbf{v}) = (cT)(\mathbf{u}) + (cT)(\mathbf{v}))

  2. Homogeneity: ((cT)(a\mathbf{v}) = c \cdot T(a\mathbf{v}) = c(aT(\mathbf{v})) = (ca)T(\mathbf{v}) = a(cT(\mathbf{v})) = a(cT)(\mathbf{v}))

Both linearity conditions are satisfied.

Consequence: The set of all linear transformations from (V) to (W), denoted (\mathcal{L}(V, W)), is itself a vector space under pointwise addition and scalar multiplication.