Understanding Existence and Uniqueness

  • #1

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.
 

Answers and Replies

  • #2
PeroK
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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##?
 

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