Linear transformations as a vector space

Notes

If we have a linear transformation $T:V \to W$ we can create a new linear transformation by multiplying it by a scalar: $U = \alpha T$. Also if we have a pair of linear transformations $T_1$ and $T_2$ with the same domain and target $U$ and $W$, the sum defines a new linear transformation $T = T_1 + T_2$ with $(T_1 + T_2)\mathbf{v} = T_1\mathbf{v} + T_2\mathbf{v}_2$.

$$ \begin{aligned} U(\alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2) &= (\alpha T)(\alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2) \\ &= \alpha(T(\alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2)) \\ &= \alpha(\alpha_1T(\mathbf{v}_1) + \alpha_2T(\mathbf{v}_2)) \\ &= \alpha\alpha_1 T(\mathbf{v}_1) + \alpha\alpha_2 T(\mathbf{v}_2) \\ &= \alpha_1\alpha T(\mathbf{v}_1) + \alpha_2\alpha T(\mathbf{v}_2) \\ &= \alpha_1(\alpha T(\mathbf{v}_1)) + \alpha_2(\alpha T(\mathbf{v}_2)) \\ &= \alpha_1U(\mathbf{v}_1) + \alpha_2U(\mathbf{v}_2) \end{aligned} $$$$ \begin{aligned} T(\alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2) &= (T_1 + T_2)(\alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2) \\ &= T_1(\alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2) + T_2(\alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2) \\ &= T_1(\alpha_1\mathbf{v}_1) + T_1(\alpha_2\mathbf{v}_2) + T_2(\alpha_1\mathbf{v}_1) + T_2(\alpha_2\mathbf{v}_2) \\ &= \alpha_1T_1(\mathbf{v}_1) + \alpha_2T_1(\mathbf{v}_2) + \alpha_1T_2(\mathbf{v}_1) + \alpha_2T_2(\mathbf{v}_2) \\ &= \alpha_1(T_1(\mathbf{v}_1) + T_2(\mathbf{v}_1)) + \alpha_2(T_1(\mathbf{v}_2) + T_2(\mathbf{v}_2)) \\ &= \alpha_1((T_1 + T_2)(\mathbf{v}_1)) + \alpha_2((T_1 +T_2)(\mathbf{v}_2)) \\ &= \alpha_1T(\mathbf{v}_1) + \alpha_2T(\mathbf{v}_2) \end{aligned} $$

Fixing the vector spaces $V$ and $W$, the collection of all linear transformations from $V$ to $W$ is denoted $\mathcal{L}(V, W)$. The set of linear transformations forms a vector space under the operations of addition and scalar multiplication. Because each linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be represented with a $m \times n$ matrix, the vector space $\mathcal{L}(\mathbb{R}^n, \mathbb{R}^m)$ is represented by the set of $m \times n$ matrices.