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Complex Numbers and Vector Space

Exercise 1 : Complex Number Inverse

Suppose $a $ and $b $ are real numbers, not both 0. Find real numbers $c $ and $d $ such that

$$ \frac{1}{a + bi} = c + di. $$

Show that

$$ \frac{-1 + \sqrt{3}i}{2} $$

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

Find two distinct square roots of $i $ (i.e., find two different complex numbers $z $ such that $z^2 = i $).

Prove the following properties and name them:

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

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

$$ (4, -3, 1, 7) + 2x = (5, 9, -6, 8). $$

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

$$ \lambda(2 - 3i, 5 + 4i, -6 + 7i) = (12 - 5i, 7 + 22i, -32 - 9i). $$

Prove the following:

Show that $(x + y) + z = x + (y + z) $ for all $x, y, z \in \mathbb{F}^n $.
Show that $(ab)x = a(bx) $ for all $x \in \mathbb{F}^n $ and all $a, b \in \mathbb{F} $.
Show that $1x = x $ for all $x \in \mathbb{F}^n $, where 1 is the multiplicative identity in $\mathbb{F} $.
Show that $\lambda(x + y) = \lambda x + \lambda y $ for all $\lambda \in \mathbb{F} $ and all $x, y \in \mathbb{F}^n $.
Show that $(a + b)x = ax + bx $ for all $a, b \in \mathbb{F} $ and all $x \in \mathbb{F}^n $.