I have trouble understanding the book Tensors and manifolds of Robert H.Wasserman. I do not know if it happens to all the beginers in algebra but I cannot understand the algebra in the proofs provided. Perhaps there is some good more elementary book explaining how these operations are performed. As it often happens in my questions, the delta kronecker is involved.
For instance, we are given a theorem which state that given bases $ {e_i} $ and $ {E_j} $ and finite-dimensional vector spaces V and W. Then the elements $ {e^{i}_j} $ of the linear maps from V to W,L(V,W),defined by $ {e^{i}_j} $: $ v —>v^{i}E_{j} $ form a basis for L(V,W).
The proof goes as follows, the definition of $ {e^{i}_j} $ the author exposes is equivalent to $ {e_k}->δ^{i}_{j}E_{j} $. I do not understand this, how can you switch between so different things? In one map you start with a vector (with no index) an in the another, with a basis vector. As I understood, the delta kronecker happenes to coincide with ε(e) where ε is the basis of the dual space of the space where is its basis. I can see indeed that the indices cancel so they give the indices belonging to e, but I could also come up with other maps which also give e with the indices i and j. Again I do not see any logic behind this. Am I missing something?
The map $e_j^i$ maps $e_i$ in $V$ to $E_j$ in $W$ and all other vectors in the basis of $V$ to 0. Using the Kronecker delta we can express this as:
$e_j^i (e_k) = \delta^i_k E_j$
If we have a general vector $v$ in $V$ with components $\{v^1, v^2, \dots \}$ relative to basis $\{e_1, e_2, \dots \}$ then we can use the Einstein summation convention to say $v=v^ke_k$. Then $e^i_j$ maps $v$ as follows:
$e^i_j(v)=e^i_j(v^ke_k)=v^k e^i_j(e_k)=v^k \delta^i_k E_j = v^i E_j$
If we now build a general linear map $A=A^j_i e^i_j$ from the basis maps $e^i_j$ then $A$ maps $v$ as follows:
$A(v)=A^j_i e^i_j(v) = A^j_i v^i E_j$
So if $A(v)$ has components $\{w^1, w^2, \dots \}$ relative to basis $\{E_1, E_2, \dots \}$ we have shown that
$w^j=A^j_i v^i$
In other words, the components $A^i_j$ of $A$ relative to basis $e^i_j$ act like the entries in a matrix that maps the components of $v$ relative to basis $e_i$ to the components of $w=A(v)$ relative to basis $E_j$. From this point of view, each map $e^i_j$ can be represented as a matrix with a 1 at positions $(i,j)$ and 0s everywhere else.