/sci/ - Science & Math

>>11440244
You need to, at the absolute least, construct the obvious bijection.
>>11440684
For [math]\mathbb{R} ^n[/math] with [math]n[/math] odd, the characteristic polynomial has odd degree, and hence at least one real root (which is an eigenvalue).
For even degree, consider [math]\mathbb{C}^n[/math] as a real vector space and the [math]J[/math] operator of multiplication by [math]i[/math].

>>11376785
Let [math](X, d_X)[/math] and [math](Y, d_Y)[/math] be two metric spaces.
We claim that [math](X \times Y, max(d_{X \times Y})[/math] is also a metric space, where [math]d_{X \times Y} ((a, b), (c, d)) = max (d_X(a, c); d_Y (b, d)[/math].
Indeed, we have that [math]d_{X \times Y} ((a, b), (e, f) = max( d_X (a, e); d_Y (b, f) ) \leq max (d_X (a, c) +d_X (c, e); d_Y (b, d) + d_Y (d, f) ) \leq max (d_X (a, c); d_Y (b, d)) + max (d_X (c, e) ; d_Y (d, f))[/math].
Proving the other parts of the definition of a metric is left as an exercise to the reader.
>>11377362
HONEY
CAN I LEND HIM MINE?
>>11376922
The one who looks smarter.

>>11372292
Because you can construct all sorts of garbage intuition out of second hand explanations and diagrams, but proper intuition about how a mathematical object actually behaves comes from its construction, its theorems, their proofs and the object's pathologies.
You memorize the construction of a formal object, form an immediate intuition about it, and gradually rework and expand it based on results and examples. This is genuinely the only sensible way of doing maths.

>>11365269
This was me, BTW.
>>11365272
Use Bhaskara.