Geometric intuition behind subspaces in $\mathbb C^n$ While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs.
After meeting complexification, I've realized that my old intuition doesn't seem to work in $\mathbb C^n$. For example, if $U$ is a subspace of $\mathbb C^n$, what is $\bar U$? In what relation is it to $U$? Etc.
Is there a handy way you think about $\mathbb C^n$?
 A: In $\mathbb{R}^n$, a vector $v$ can be thought of as a list of components $v_i$.  Each component has a magnitude, which is a positive real number $|v_i|$, and a direction, which is one of $1$ and $-1$.
In $\mathbb{C}^n$, a vector $v$ is likewise a list of components $v_i$.  Each component has a magnitude $|v_i|$, and a direction, which now can be any angle between $0$ and $2\pi$ and not just positive and negative directions.
In a sense, this means an element of $\mathbb{C}^n$ is a vector of vectors.


*

*The norm of a vector in $\mathbb{C}^n$ is just like the norm of a vector in $\mathbb{R}^n$: you add up the magnitude squared of each component, and take the square root.  Just like in $\mathbb{R}^n$ where you ignore the angle $1$ or $-1$ in computing the norm, here you ignore the argument of the complex number. 

*The conjugate of a vector is formed by reversing the angle in each component.

*Multiplying by a complex number $re^{i \theta}$ scales the vector by a certain amount $r$, and rotates each component by some angle $\theta$.  This is the same as in the real case, where multiplication by $2$ and $-2$ are essentially the same up to changing the sign of the components.
This is how I think of it anyway.  Hope you find this helpful.
