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About the linear dependence of linear operators

  1. Jun 20, 2009 #1
    Notations:
    F denotes a field
    V denotes a vector space over F
    L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is composition of functions.
    τ denotes a linear operator contained in L(V)
    ι denotes the multiplicative identity of L(V)

    Question:
    Why is n2 + 1 vectors:
    ι, τ, τ2, ... ,τn2
    are linearly dependent in L(V)?

    I wonder why, if these vectors are linear dependent, then one of them can be expressed as the linear combination of other vectors, but how?

    Thanks for any help!
     
  2. jcsd
  3. Jun 27, 2009 #2

    morphism

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    Science Advisor
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    I presume V is n-dimensional? If this is the case, then your list of vectors is linearly dependent because dim L(V) = ____ (fill in the blank).
     
  4. Jun 28, 2009 #3
    Yes, V is n-dimensional.

    dim L(V) = n2

    Now I understand, if n2 + 1 vectors:
    ι, τ, τ2, ... ,τn2
    are llinear independent, then these vectors span L(V) and dim L(V) = n2 + 1 ≠ n2 = dim L(V), it's a contradiction, so, these vectors must be linear dependent.

    morphism, thanks a lot!
     
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