Let \(\mathcal{U}, \mathcal{V}\) be vector spaces over the field \(\mathbb{F}.\) A map \(T: \mathcal{U} \rightarrow \mathcal{V}\) is a linear transformation \(\forall \, \underline{u},\underline{u}_1,\underline{u}_2 \in \mathcal{U}\) and \(\forall \, \lambda \in \mathbb{F}\) if it satisfies:

\[T(\underline{u}_1+\underline{u}_2)=T(\underline{u}_1)+T(\underline{u}_2)\] \[T(\lambda \underline{u}) = \lambda T(\underline{u}).\]

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

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

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

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

The rank of \(T\) is the dimension of the image of \(T:\)

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

The nullity of \(T\) is the dimension of the kernel of \(T:\)

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

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

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