Matrix representation

The matrix representation of a linear map $T: V \to W$ with respect to bases $\mathcal{B} = \{v_1, \ldots, v_n\}$ of $V$ and $\mathcal{C} = \{w_1, \ldots, w_m\}$ of $W$ is the $m \times n$ matrix $[T]_{\mathcal{B}}^{\mathcal{C}}$ whose columns are the coordinate vectors of $T(v_j)$ in the basis $\mathcal{C}$.

That is, if $T(v_j) = \sum_{i=1}^m a_{ij} w_i$, then the $(i,j)$ entry of the matrix is $a_{ij}$.

Properties

• Composition: $[ST]_{\mathcal{B}}^{\mathcal{D}} = [S]_{\mathcal{C}}^{\mathcal{D}} [T]_{\mathcal{B}}^{\mathcal{C}}$.
• Change of basis: If $P$ is the change-of-basis matrix from $\mathcal{B}$ to $\mathcal{B}'$, then $[T]_{\mathcal{B}'}^{\mathcal{C}'} = Q^{-1} [T]_{\mathcal{B}}^{\mathcal{C}} P$.
• Similarity: Two matrices represent the same operator (different bases) iff they are similar.

Standard representation

For $T: \mathbb{R}^n \to \mathbb{R}^m$, the standard matrix is $[T(e_1) \mid \cdots \mid T(e_n)]$.

Invariants

The rank , determinant (for square matrices), and eigenvalues are independent of the basis choice.