Past exam of the mathematical methods I course of the University of Oxford / 2021 / 8 / a

The linear span is $\{\lambda \mathbf{a}+\mu \mathbf{b}:\lambda ,\mu \in \mathbb{R}\}$, a vector subspace of ${\mathbb{R}}^3$. Its dimension is zero if both vectors vanish, one if at least one is nonzero and they are linearly dependent, and two if they are linearly independent. The corresponding geometric sets are the origin, a line through the origin, and a plane through the origin.

Contract the Levi-Civita symbol identity ${\epsilon }_{ijk}{\epsilon }_{i\ell m}={\delta }_{j\ell }{\delta }_{km}-{\delta }_{jm}{\delta }_{k\ell }$. Using Einstein summation convention,This Lagrange identity for the cross product includes zero and parallel vectors without a separate division by their norms.