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Vector Space Basis

  1. Dec 2, 2009 #1
    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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  3. Dec 3, 2009 #2

    HallsofIvy

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    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".
     
  4. Dec 3, 2009 #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?
     
  5. Dec 3, 2009 #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").
     
    Last edited: Dec 3, 2009
  6. Dec 4, 2009 #5

    HallsofIvy

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    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]
     
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