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The set of linear transformations between two vector spaces is a vector space.

Created over 8 years ago, updated 10 days ago

Theorem: The set of all linear transformations from a vector space (V) to a vector space (W) (over the same field (\mathbb{F})), denoted (\mathcal{L}(V, W)) or ( ext{Hom}(V, W)), is itself a vector space over (\mathbb{F}).

Operations:

  • Addition: ((S + T)(\mathbf{v}) = S(\mathbf{v}) + T(\mathbf{v}))
  • Scalar multiplication: ((cT)(\mathbf{v}) = c \cdot T(\mathbf{v}))

Zero vector: The zero transformation (Z(\mathbf{v}) = \mathbf{0}) for all (\mathbf{v}).

Verification: The sum and scalar multiple of linear transformations are themselves linear transformations.

Dimension: If (\dim V = n) and (\dim W = m), then (\dim \mathcal{L}(V, W) = mn). This is isomorphic to (M_{m imes n}(\mathbb{F})) via the matrix representation map.