Let $B=\{ v_i \}_{i=1}^n$ base of V, and $T:V\rightarrow V$ a linear operator, while $T(v_1)=\sum_{i=1}^nv_i$. Find the sum of $[T^{-1}]_B$ elements

Question

I've came across the question described in the title, and got stuck. The first step I thought about is to find the matrix which represents $T$, by looking at the image of each vector in the base, but I have no idea how to find the image of the vector except $v_1$:

$T(v_1)=(1,1,1,......,1)$

$T(v_i)=?$

Answer 1

The question does not give you enough information to completely determine the linear transformation (or the matrix that represents it). So, "finding the matrix which represents $T$" won't work.

Hint: Let $x$ denote the (column-) vector $x=(1,1,\dots,1)$. Note that $[T^{-1}]_B x$ produces a vector whose entries are the sum of each row of $[T^{-1}]_B$.