Understanding Existence and Uniqueness

[Bishamonten]

Homework Statement

Show that for every α ∈ ℂ with α ≠ 0, there exists a unique β ∈ ℂ such that αβ = 1

Homework Equations

Definition[/B]: $\mathbb {F^n}$

$\mathbb {F^n}$ is the set of all lists of length n of elements of $\mathbb {F}$ :
$\mathbb {F}$ = {$(x_1,...,x_n) : x_j ∈ \mathbb {F} for j = 1,...,n$}

Definition: addition in $\mathbb {F^n}$
$(x_1,...,x_n) + (y_1,...,y_n) = (x_1 + y_1,..., x_n + y_n)$

Definition: scalar multiplication in $\mathbb {F^n}$
$λ(x_1,...,x_n) = (λx_1,...,λx_n)$
$λ ∈ \mathbb {F}, (x_1,...,x_n) ∈ \mathbb {F^n}$

And the 7 other properties of fields: https://en.wikipedia.org/wiki/Field_(mathematics)#Classic_definition

The Attempt at a Solution

$α ∈ ℂ → α = a + ui$ $a, u ∈ ℝ$
$β ∈ ℂ → β = b + vi$ $b, v ∈ ℝ$
$∃γ∈ℂ$ such that $γ = \frac{1}{α} = \frac {1}{a+ui} = c + di$ $c,d ∈ ℝ$

Proof:

Existence

By multiplicative identity: $$ \frac {1}{a+ui}\frac {(a-ui)}{(a-ui)} = \frac {(a-ui)}{(a^2 + u^2)} = \frac {(a)}{(a^2 + u^2)} +\frac {(-u)}{(a^2 + u^2)}i $$

By definition of real numbers:
$$ = s + ti$$ $$s, t ∈ ℝ $$
By definition of complex numbers:
$$ = ψ ∈ ℂ $$

Let $b = \frac {a}{a^2 + u^2}, v = \frac {-u}{a^2 + u^2}$, then

By substitution: $$ β = b + vi = \frac {a}{a^2 + u^2} + \frac {(-u)}{a^2 + u^2}i = \frac {1}{a+ui}$$

$$ → αβ = (a+ui)\frac {1}{(a+ui)} = 1$$

Uniqueness

Suppose ∃δ∈ℂ such that αδ = 1

$$ αβ = 1 $$ and $$ αδ = 1 $$

By transitivity of equality:
$$αβ = αδ $$
By cancellation:
$$ β = δ $$My question is, have I correctly proved the uniqueness part? Was I also doing a bit too much with the existence portion of the proof? This very simple exercise comes from Axler's Linear Algebra Done Right. I always felt a bit iffy proving uniqueness in linear algebra, and wish to brush up the subject again.

 

[PeroK]

I'm not sure this is correct. If you assume the field axioms for $\mathbb{C}$, then you have a unique inverse for any non-zero element, which is what you are trying to prove.

I suggest you can assume the field axioms for $\mathbb{R}$ and what you have proved already for $\mathbb{C}$ - commutivity, associativity, distributive law etc. But, I don't know exactly how your book approaches this.

For the uniqueness part, I'm not sure you can invoke a "cancellation" law. But, once you have proved existence, you might think about how you show uniqueness. Hint: similar to cancellation but using what you have shown already in this proof.

Finally, you took a lot of algebra to show that $\frac{a - bi}{a^2 + b^2} (a+bi) = 1$. You could simply have shown that directly and thereby proved existence. Note that you need to say something about why $\frac{a - bi}{a^2 + b^2}$ is well-defined. Hint: why does this not apply to $0$?