Let U,V\mathcal{U}, \mathcal{V} be vector spaces over the field F.\mathbb{F}. A map T:U→VT: \mathcal{U} \rightarrow \mathcal{V} is a linear transformation ∀ u‾,u‾1,u‾2∈U\forall \, \underline{u},\underline{u}_1,\underline{u}_2 \in \mathcal{U} and ∀ λ∈F\forall \, \lambda \in \mathbb{F} if it satisfies:

T(u‾1+u‾2)=T(u‾1)+T(u‾2)T(\underline{u}_1+\underline{u}_2)=T(\underline{u}_1)+T(\underline{u}_2) T(λu‾)=λT(u‾).T(\lambda \underline{u}) = \lambda T(\underline{u}).

Let U,V\mathcal{U}, \mathcal{V} be vector spaces over the field F\mathbb{F} and T:U→VT: \mathcal{U} \rightarrow \mathcal{V} be a linear transformation. The image of TT is the set of vectors mapped by TT from U:\mathcal{U}:

Im T:={T(u‾):u‾∈U}\mathrm{Im} \, T := \{T(\underline{u}):\underline{u}\in\mathcal{U}\}

The kernel or null space of TT is the set of vectors in U\mathcal{U} which are mapped to 0‾\underline{0} in V\mathcal{V} by TT

ker T:={u‾∈U:T(u‾)=0‾}\mathrm{ker} \, T := \{\underline{u} \in \mathcal{U} : T(\underline{u}) = \underline{0}\}

The rank of TT is the dimension of the image of T:T:

rank T:=dim(Im T)\mathrm{rank} \, T := \mathrm{dim}(\mathrm{Im} \, T)

The nullity of TT is the dimension of the kernel of T:T:

null T:=dim(ker T)\mathrm{null} \, T := \mathrm{dim}(\mathrm{ker} \, T)

The rank-nullity theorem states that the dimension of U\mathcal{U} is the sum of the rank and nullity of TT if U\mathcal{U} is finite dimensional:

dim(U)=rank T+null T\mathrm{dim}(\mathcal{U}) = \mathrm{rank}\, T + \mathrm{null} \, T