About the linear dependence of linear operators

[sanctifier]

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!

 

[morphism]

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).

 

[sanctifier]

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!