$\mathbb{R}^n$ and $\mathbb{C}^n$

Show that $\alpha +\beta = \beta +\alpha$ for all $\alpha , \beta \in \mathbb {C}$.

Show that $(\alpha +\beta )+\lambda = \alpha +(\beta +\lambda )$ for all $\alpha , \beta , \lambda \in \mathbb {C}$.

Show that $(\alpha \beta ) \lambda = \alpha (\beta \lambda )$ for all $\alpha , \beta , \lambda \in \mathbb {C}$.

Show that $\lambda (\alpha +\beta ) = \lambda \alpha +\lambda \beta$ for all $\lambda , \alpha , \beta \in \mathbb {C}$.

Show that for every $\alpha \in \mathbb {C}$, there exists a unique $\beta \in \mathbb {C}$ such that $\alpha +\beta = 0$.

Show that for every $\alpha \in \mathbb {C}$ with $\alpha \neq 0$, there exists a unique $\beta \in \mathbb {C}$ such that $\alpha \beta = 1$.

Show that

is a cube root of 1 (meaning that its cube equals 1).

Find two distinct square roots of $i$.

Find $x \in \mathbb {R}^{4}$ such that

Explain why there does not exist $\lambda \in \mathbb {C}$ such that

Show that $(x+y)+z = x+(y+z)$ for all $x, y, z \in \mathbf{F}^{n}$.

Show that $(a b) x = a(b x)$ for all $x \in \mathbf{F}^{n}$ and all $a, b \in \mathbf{F}$.

Show that $1 x = x$ for all $x \in \mathbf{F}^{n}$.

Show that $\lambda (x+y) = \lambda x+\lambda y$ for all $\lambda \in \mathbf{F}$ and all $x, y \in \mathbf{F}^{n}$.

Show that $(a+b) x = a x+b x$ for all $a, b \in \mathbf{F}$ and all $x \in \mathbf{F}^{n}$.