Problem
(a) Show that if we think of $\mathbb{C}$ as a vector space over $\mathbb{R}$, then the list $(1 + i, 1 - i)$ linearly independent.
(b) Show that if we think of $\mathbb{C}$ as a vector space over $\mathbb{C}$, then the list $(1 + i, 1 - i)$ linearly dependent.
Solution
(a) if we let $x,y \in \mathbb{R}$, then $$0 + 0i = 0 = x(1 + i) + y(1 - i) \stackrel{P1A6 \land D1.1}{=} x + y + (x - y)i$$ By equating coefficients we see that $x + y = x - y = 0$, from which it follows that $x = y = 0$, thus $(1 + i, 1 - i)$ is linearly independent.
(b) if we let $x,y \in \mathbb{C}$ we see that $x = i$, $y = 1$ is a non-trivial linear combination of the given vectors that equate to $0$: $$x(1 + i) + y(1 - i) = i - 1 + 1 - i = 0$$ Therefore, by definition 2.19, the given list is linearly dependent.