題組內容

2. Let $L: \mathbf{R}^2 \to \mathbf{R}^3$ be $L(\mathbf{x}) = (x2, x1 - x2, x2 - x_1)^\mathrm{T}$.

(b) (10%) Give the following two ordered bases: \( E = \{\mathbf{u}1, \mathbf{u}2\} \) where \( \mathbf{u}1 = (2, 1)^\mathrm{T} \), \( \mathbf{u}2 = (1, 3)^\mathrm{T} \), and \( F = \{\mathbf{v}1, \mathbf{v}2, \mathbf{v}3\} \) where \( \mathbf{v}1 = (1, 0, 0)^\mathrm{T} \), \( \mathbf{v}2 = (1, 1, 0)^\mathrm{T} \), \( \mathbf{v}3 = (1, 1, 1)^\mathrm{T} \). Find the matrix representation of \( L \) with respect to the ordered bases \( E \) and \( F \).