Theorem: If \(T: V \to W\) is a linear transformation and \(c\) is a scalar, then \(cT\) is also a linear transformation.
Proof:
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Additivity: \((cT)(\mathbf{u} + \mathbf{v}) = c \cdot T(\mathbf{u} + \mathbf{v}) = c(T(\mathbf{u}) + T(\mathbf{v})) = cT(\mathbf{u}) + cT(\mathbf{v}) = (cT)(\mathbf{u}) + (cT)(\mathbf{v})\)
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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.
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