In ${\bf R}$, the length of a number $x$ (better knows as its absolute value) is $\sqrt{x^2}$. In ${\bf R}^n$, then length $|{\bf x}|$ of a vector is $\sqrt{\langle {\bf x} | {\bf x} \rangle}$. We would like something similar to apply in ${\bf C}^n$, but first we have to understand what length means. That's the subject of the first video:

The standard inner product on ${\bf C}^n$ is $$\langle {\bf x} | {\bf y} \rangle = \overline{x^T} y = \sum_{j=1}^n \bar x_j y_j.$$ Don't forget to take complex conjugates! 90% of all mistakes that M346 students make on this subject come from forgeting to conjugate the elements of ${\bf x}$ when computing $\langle {\bf x} | {\bf y} \rangle$. Also,

• This is not bilinear! Instead, it is linear in the second factor, but is conjugate-linear (sometimes called antilinear) in the first factor: $\langle {\bf x}| c{\bf y} \rangle = c \langle {\bf x}| {\bf y} \rangle$, but $\langle c {\bf x}| {\bf y} \rangle = \bar c \langle {\bf x}| {\bf y} \rangle$, where $c$ is a (possibly complex) scalar.
• Order matters. $\langle {\bf x} | {\bf y} \rangle$ is the complex conjugate of $\langle {\bf y} | {\bf x} \rangle$.
• The inner product is positive, in that $\langle {\bf x} | {\bf x} \rangle = |{\bf x}|^2 > 0$.

In the second video, we use these properties to define axioms for complex inner products, and we look at a few examples

The Schwarz inequality still works. You just have to adjust the proof a little. However, since $\langle {\bf x} | {\bf y} \rangle$ is a complex number, it doesn't make sense to talk about the angle between ${\bf x}$ and ${\bf y}$.

There are complex versions of all of our favorite real examples of vector spaces. Given any real vector space $V$, we can make a complex vector space $V_c$ by considering combinations $a {\bf x} + i b {\bf y}$. For instance, $C^n$ is the complexification of $R^n$ and $C[t]$ is the complexification of $R[t]$. This doesn't account for all complex vector spaces, but most of the common ones are complexifications of real vector spaces. We can also build a complex inner product on $V_c$ from a real inner product on $V$. (See problem 8)