The set of linear transformations between two vector spaces is a vector space.
Created almost 9 years ago, updated 5 months ago
ID24112
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}).
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.
**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.