Prof. Alejandro Ucan-Puc
Definition: Let $B = \{v_1, v_2, \dots, v_n\}$ be an ordered basis for $V$. For any $v \in V$, there exists a unique representation:
$$v = c_1 v_1 + c_2 v_2 + \dots + c_n v_n$$The column vector $[v]_B = \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{pmatrix} \in \mathbb{R}^n$ is called the coordinate vector of $v$ relative to $B$.
Definition: Let $T: V \to W$ be a linear transformation, $B = \{v_1, \dots, v_n\}$ a basis of $V$, and $C = \{w_1, \dots, w_m\}$ a basis of $W$.
The matrix associated to $T$ with respect to $B$ and $C$ is the $m \times n$ matrix:
$$[T]_B^C = \begin{pmatrix} [T(v_1)]_C & [T(v_2)]_C & \dots & [T(v_n)]_C \end{pmatrix}$$For any vector $v \in V$, the image coordinate vector $[T(v)]_C$ is obtained via matrix multiplication:
$$[T(v)]_C = [T]_B^C \, [v]_B$$This converts abstract linear operations into computational matrix multiplications.
Let $T: \mathbb{R}^2 \to \mathbb{R}^3$ be defined by:
$$T(x,y) = (x+y, \, x-y, \, 2x+3y)$$Find the matrix $[T]_B^C$ with respect to standard canonical bases $B$ and $C$.
$T(1,0) = (1, 1, 2)^T \implies [T(1,0)]_C = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$
$T(0,1) = (1, -1, 3)^T \implies [T(0,1)]_C = \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}$
Hence: $[T]_B^C = \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ 2 & 3 \end{pmatrix}$.
Matrix representations allow computing linear operations directly via matrix algebra.
Next Session: Eigenvalues, Eigenvectors, and Stationary States.