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which is fine. Then it says that [tex]A(\textbf{e}_{i})[/tex] is a vector, given by:[tex]Let \textbf{p}=\sum_{i}p_{i}\widehat{\textbf{e}}_{i}[/tex]

[tex]A(\textbf{p}) = \sum_{i}p_{i}A(\textbf{e}_{i})[/tex]

[tex]A(e_{i}) = \sum_{j}A_{j}(p_{i})e_{j} = \sum_{j}A_{ji}e_{j}[/tex].

The fact that its a vector is fine with me, but I can't get my head around the equation for it. why does the operator acting on one of the base vectors depend on [tex]p_{i}[/tex]? Surely the base vectors are independent of [tex]p_{i}[/tex] and so should be any operation acting on them.