Vector Space Basis: Standard or Odd?

[Newtime]

In short: does every vector space have a "standard" basis in the sense as it is usually defined i.e. the set {(0,1),(1,0)} for R²? And another example is the standard basis for P³ which is the set {1,t,t²}. But for more abstract or odd vector spaces such as the space of linear transformations (automorphisms?) what would the standard basis be?

 

[HallsofIvy]

No, not every vector space has a "standard" basis because there are many vector spaces no one has every looked at! There are, after all, an infinite number of vector spaces! A "standard" basis is simply a basis that has been declared "standard".

 

[Newtime]

This makes sense. But a related question: What then would be any basis for the space of linear transformations of R^2 onto R^2? Any 2x2 matrix? or perhaps 4 arbitrary 2x2 matrices?

 

[JSuarez]

The space of Linear transformations L(U,V), where U and V are finite dimensional linear spaces, with dimensions m and n, is itself a linear space with dimension mn; its "standard" basis is the set of matrices Ekl, defined by:

$$[Ekl]_{ij}$$ = $$\delta_{kilj}$$

These basis are called "standard", because they are built using only the unit (1) of the scalar field; therefore, given a representation of the vector relative to this basis, its coordinates are, in a sense, immediate.

Regarding the general question, every vector space, finite or infinite dimensional, has indeed a basis of this type, called an Hamel basis, and also because they are completely classified by their dimension (vector spaces with the same dimension are isomorphic); of course, in infinite dimensional spaces, the Hamel basis is uncomputable (and unenumerable); in finite dimensions, it coincides with the usual canonical (or "standard basis").

 

[HallsofIvy]

The "standard basis" for the vector space of 2 by 2 matrices (while not every vector space has a "standard" basis, simple one like this do) consists of the four matrices
$$\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$$
$$\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}$$
$$\begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}$$
and
$$\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$$

So that any matrix can be written as
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}= a\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}+ b\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}+ c\begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}+ d\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}$$