Complex 3-D Euclidean space - inner product

Question

1st question: Lets say we have a 3-D complex euclidean space. How do we geometrically draw this space? if 3-D real Euclidean space is represented by these base vectors:

2nd question: Is there a proof that for the scalar product in a 3-D complex Euclidean space and written in Dirac notation it holds this:

$$\langle A | B\rangle = {A_x}^*B_x + {A_y}^*B_y + {A_z}^* B_z $$

where ${A_x}^*$ is a conjugate of $A_x$. This equation can be found on Wikipedia.

Answer 1

A complex number can be represented by a pair of real numbers with some additional rules for multiplication. Therefore, a complex space of dimension $n$ can be seen as a real space of dimension $2n$, with some additional structure of multiplication (which we may ignore when it's not needed: for example, when drawing).

• It's easy to draw a 1-dimensional complex space, because it is also a 2-dimensional real space, a plane.
• It's next to impossible to accurately draw a 2-dimensional complex space, because it is also a 4-dimensional real space, and humans are not good at visualizing or drawing 4 dimensions.
• Since the 3-dimensional complex space is also a 6-dimensional real space, it is much too large for us to draw.

So, people resort to surrogates, such as Reinhardt diagrams (seen here), or drawing slices of $\mathbb C^n$. These don't represent the space, just some aspects of it.

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For the second question: before proving something about a mathematical object, one must define it. How do you define the scalar product on $\mathbb C^3$? Two definitions are I know are: $\langle a,b\rangle=\sum \bar a_j b_j$ and $\langle a,b\rangle=\sum a_j \bar b_j$. These are not the same; some people prefer the first and others prefer the second. There's no accounting for taste.