I have a vague understanding of the concepts mentioned in the title. But have some questions to clear things up:
Firstly, what does this mean on Wikipedia?
For $\mathbb{R}^3$ the set of vectors $\{e_1,e_2,e_3\}$ is called the standard basis and forms and orthonormal basis of $\mathbb{R}^3$ with respect to the standard dot product. Note that both the standard basis and standard dot product rely on viewing $\mathbb{R}^3$ as the Cartesian product $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$
In italic are the things that I dont understand. What does it mean "in respect to the standard dot product", can you be an orthonormal basis with respect to a different inner product? If so, please give an example, and if that's not the case, what does this statement mean?
Also, what does it mean by "relies heavily on viewing $\mathbb{R}^3$ as the cartesian product"? How else could you possible view it?
Finally, what is a "cannonical inner product", It is mentioned on one of my linear algebra books, and is basically just the dot product, except it's not commutative since you take the complex conjugate of each component the first vector.
While also answering these questions, I'd like to know what an Inner-product actually is, and what is the motivation for developing them? Thank you
Every basis $(e_1,\dots,e_n)$ of an $n$-dimensional vector space $V$ over $\Bbb R$ is orthonormal with respect to a (unique) inner product $\langle~,~\rangle$ on $V$, defined by $$\left\langle\sum_{i=1}^nx_ie_i,\sum_{j=1}^ny_je_j \right\rangle=\sum_{i=1}^nx_iy_i. $$ When $V$ has a canonical basis $\mathcal B$, the canonical inner product on $V$ is the one for which $\mathcal B$ is orthonormal.
As commented by @IzaakvanDongen, not every finite dimensional vector space over some field $K$ has a canonical basis, but $K^n$ does, and that basis is tied to the cartesian product structure of this vector space.
Finally, the definition of inner products on complex vector spaces differs from the definition of inner products on real vector spaces, the only difference being that $\langle~,~\rangle$ is required to be antilinear w.r.t. its first argument, instead of linear (some authors choose the other convention: antilinearity w.r.t. the second argument).