Can someone explain to me in layman's terms the difference between a dot product and inner product?

Question

I am reading this and it's not exactly clearing things up for me: https://math.stackexchange.com/questions/476738/difference-between-dot-product-and-inner-product

The only thing I got from searching the web so for is that dot product and inner product are basically the same thing except dot product only works on real numbers while inner product can work on any type of number. Is this correct?

I am currently reading about SVMs and hoping to go into neural networks, and am stuck at kernel method/trick.

Answer 1

A very simple answer: those are synonyms. However (as hinted at in the math SE post), there are shades of meaning.

• the dot product is usually used in a very concrete sense, it is the standard inner product $(x_1,y_1) \cdot (x_2,y_2)$ defined in high school for the plane as $(x_1,y_1) \cdot (x_2,y_2) = x_1 x_2 + y_1 y_2$, and its extension to 3D space and also to $\mathbb{R}^n$.

• while inner product is used more generally, on vector spaces other than $\mathbb{R}^n$, for instance.

Answer 2

"Dot product" is a special case of "inner product". We define "dot product" on spaces $\mathbb R^n$. But "inner product" is also defined on other spaces; and indeed there are other inner products on $\mathbb R^n$ in addition to the dot product. See the Wikipedia page for more information.