Let us compute the determinant according to (4.7) in a 2-D space and compare it to the classical determinant. Take a basis {b1, b2}, not necessarily orthonormal. Let the linear mapping f be such that f[b1] = x and f[b2] = y. Develop x and y onto the basis x = x1b1 + x2b2 and y = y1b1 + y2b2. Use I2 = x ∧ y and compute the determinant according to (4.7). Now compute the matrix of f on the given basis, and compute its classical determinant. The results should match.