# Definition Given two [[Vector Space|vector spaces]] $V$ and $W$ over a [[field|field]] $\mathbb{F}$, a *linear [[map]]* $f:V\rightarrow W$ is one that satisfies $f(a_1 \pmb{v_1} + a_2 \pmb{v}_2) = a_1 f(\pmb{v}_1) + a_2 f(\pmb{v}_2)$ for all $_1, a_2 \in \mathbb{F}$ and all $\pmb{v}_1, \pmb{v}_2 \in V$. # Rank-Nullity Theorem Note that the [[Domain, Co-domain, Image|image]] ($\text{im}(f)$) is a [[vector subspace]] of $W$ while the [[Kernel of a Map|kernel]] ($\text{ker}(f)$) are [[vector subspace|vector subspaces]] of $V$. The dimension of the image is known as the *rank* of $f$ and the dimension of the kernel is known as the *nullity* of $f$. The rank nullity theorem says. $ \text{dim}(V) = \text{rank}(f) + \text{nullity}(f) $ # Homomorphisms and Isomoprhisms Setting $a_1=a_2 = 1$ in the definition above, note that $f$ preserves the additive structure of the vector spaces, and thus is a *vector space [[homomorphism]]*. Specifically, it is a [[group homomorphism]] between the two additive [[group|groups]] $V$ and $W$. If $f$ is additionally a [[Types of Maps|bijection]], then it is a *[[Vector Space Isomorphism]]* (or a [[group isomorphism]] between the additive groups $V$ and $W$) It can be shown that a linear map is an isomorphism if and only if 1. $\dim(V) = \dim(W)$ 2. $f(\pmb{v}) = 0 \implies \pmb{v} = 0$