> [!definition] > > Let $V \iso \real^n$ be a [[Vector Space|vector space]], $v \in V$ be a vector, and $\seqf{e_i}$ be a [[Basis|basis]] for $V$, then $v$ has the coordinate representation > $ > v = \sum_{i = 1}^nv^ie_i > $ > In the **Einstein summation** convention, we omit the sum and simply write $v = v^ie_i$. The superscripted term usually represents a coordinate, and the subscripted term represents the basis vector. Einstein summation is identified by the presence of the same index appearing twice. > [!theorem] > > Let $V$ and $W$ be vector spaces with ordered bases $\bracs{e_i}$ and $\bracs{f_j}$, respectively. A [[Linear Transformation|linear transformation]] $T: V \to W$ can be represented in matrix form by passing it through coordinates: > $ > v \mapsto v^ie_i \mapsto v^i a_{i}^j f_j > $ > where the lower subscript represents the input and the upper subscript represents the output. > [!definition] > > Let $V_1, V_2, V_3$ be vector spaces with ordered bases $\bracs{e_i}$, $\bracs{f_j}$, and $\bracs{g_k}$, respectively. Let $S: V_1 \to V_2$ and $T: V_2 \to V_3$ with coordinate representation $(a_i^j)$ and $(b_j^k)$ respectively, then for any $v \in V$, we can represent > $ > v \mapsto v^ie_i \mapsto v^ia_i^j f_j \mapsto v^i a_i^jb_j^k g_k > $ > the matrix product $[T][S]$ has representation $(c_i^k)$ where $c_i^k = a_i^jb_j^k$.