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

**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 \(\text{Hom}(V, W)\), 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 \(M_{m \times n}(\mathbb{F})\) via the matrix representation map.Copy