Vector Spaces: canonical basis for the usual vector spaces I'm looking for a listing of the canonical basis for the most common vector spaces.
Some standard basis are not so obvious. For instance, the basis for vector space $\Bbb C^2$ is $\{ (1,0),(i,0),(0,1),(0,i)\}$ using $\Bbb R^2$ scalars, or $\{ (1,0),(0,1)\}$ if the scalars are complex.
So what I'd like to find are the canonical basis for $\Bbb R^n$, $\Bbb C^n$, vector spaces for matrices, polynomials, etc.
Any reference text, book or source will be appreciated!
 A: You confuse canonical with standard. The only vector space, up to isomorphism, that has a canonical basis is $\{0\}$, the trivial vector space. Any other vector space has bases (even infinite dimensional ones if you believe in the axiom of choice) but none of these is canonical. For different purposes different bases may be more or less suitable. There however what are called standard bases, for some vector spaces, certainly not all.  Your question seems to indicate you are already familiar with the standard bases in the most familiar vector spaces. For polynomial spaces you use the vectors of the form $x^n$. For matrices you take matrices consisting of all zeros except for one $1$. 
In general though you will find that the only vector spaces that admit a standard basis are those that admit a standard isomorphism with $\mathbb R^n$ or $\mathbb C^n$ (in the finite dimensional case).
A: I'm not sure what you mean by "canonical." Some of those vector spaces have multiple canonical bases: the one you use depends on your purpose. For example, the usual basis for real polynomials is the unitary monomials
$$\{1, x, x^2, x^3, \, \dots , \, x^n, \dots \}$$
However, if you are working with polynomial approximation you may prefer the Chebyshev polynomials
$$\{1, x, 2x^2-1, 4x^3-3x, \, \dots , \, \cos(n\arccos x), \, \dots \}$$
