Does the Equation U·U = ||U||^2 Hold for Complex Vectors?

[SoulofLoneWlf]

Homework Statement

well i am told basically this
|u|^2 = U dot U except when in using imaginary numbers
i tried using the example below and a few others but it seems to always work for me :/
but i need the process or to show this is not true

Homework Equations

||u|| = U dot U in general not complex :/

The Attempt at a Solution

2i + 5

 

[CompuChip]

What are u and U?
Are they the same (U = u)?
Is u a vector, of which |u| is the norm, or just a number of which |u| is the absolute value (for real numbers) or the modulus (for complex ones). Or is it a matrix, for which |u| is some matrix norm?

If it is a number, it's rather easy to show.
The modulus (length) of 2i + 5 is $\sqrt{2^2 + 5^2}$ which is not equal to (2i + 5)^2.

In general, the correct expression is
|u|^2 = u . u*
where u* is the complex conjugate of u.