A linear map from a vector space $V$ to $W$ is a function

$$ T : V \to W $$

with the following properties:

• Additivity - $T(u+v) = Tu + Tv$
• Homogeneity - $T(\lambda v) = \lambda T(v)$

The set of linear maps from $V$ to $W$ is denoted $\mathcal{L}(V, W)$. $\mathcal{L}(V) = \mathcal{L}(V,V)$.

Most things that you can imagine are linear maps! Zero itself is one - $0 \in \mathcal{L}(V,W)$ is defined by $0v = 0$.

<aside> 💡 Note that the $0$ on the LHS is the linear map from $V$ to $W$, while the $0$ on the RHS is the additive identity in $W$. Context matters!

</aside>

Even differentiation and integration are linear maps $D$ and $I$, as the additivity and homogeneity properties hold in the vector space of polynomials!

The linear map lemma states that:

Suppose $v_1, \dots, v_n$ is a basis of $V$ and $w_1,\dots,w_n \in W$. Then there exists a unique linear map $T : V \to W$ such that $Tv_k = w_k$ for each $k = 1,\dots,n$.

Note that $\mathcal{L}(V,W)$ is a vector space itself! You can see this if you define addition as $S+T(v) = S(v) + T(v)$. Additionally, the product of linear maps is associative.

Linear maps also have to map $0$ to $0$ - $T(0) = 0$.