Exercise 38

Let ${\mathbf x} = (x_1, x_2)$ and ${\mathbf y} = (y_1, y_2)$. Prove that ${\mathbf x}$ and ${\mathbf y}$ are linearly independent if and only if $$\det \begin{pmatrix} x_1 & y_1 \\ x_2 & y_2 \end{pmatrix} \neq 0.$$