Vector Space Basis: Standard or Odd?

In summary: You can summarize this conversation like this:Every vector space has a basis, but not every basis is "standard". For more abstract or odd vector spaces, there may be no standard basis.
  • #1
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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?
 
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  • #2
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".
 
  • #3
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?
 
  • #4
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:

[tex][Ekl]_{ij}[/tex] = [tex]\delta_{kilj}[/tex]

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").
 
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  • #5
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
[tex]\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}[/tex]
[tex]\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}[/tex]
[tex]\begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}[/tex]
and
[tex]\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}[/tex]

So that any matrix can be written as
[tex]\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}[/tex]
 

Related to Vector Space Basis: Standard or Odd?

1. What is a vector space basis?

A vector space basis is a set of linearly independent vectors that can be used to represent any vector within a given vector space. It serves as a foundation for understanding the structure and properties of vector spaces.

2. What is a standard basis?

A standard basis is a set of vectors that are used to represent points in space in a standard coordinate system. In a standard basis, the vectors are typically orthogonal (perpendicular) and have a length of 1.

3. What is an odd basis?

An odd basis is a set of vectors that are used to represent points in space in a non-standard coordinate system. Unlike a standard basis, the vectors in an odd basis may not be orthogonal or have a length of 1. They may also have a different number of dimensions.

4. How do you determine the basis for a vector space?

To determine the basis for a vector space, you can use the Gaussian elimination method to find the linearly independent vectors that span the space. Alternatively, you can use the Gram-Schmidt process to orthogonalize a set of vectors and then check for linear independence.

5. Can a vector space have more than one basis?

Yes, a vector space can have multiple bases. This is because there are often multiple ways to represent a vector within a given vector space. However, all bases for a vector space will have the same number of vectors, known as the dimension of the space.

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