Linear combinations of cash flows

Question

We consider cash flow vectors over T time periods, with a positive entry meaning a payment received, and negative meaning a payment made. A (unit) single period loan, at time period t, is the T-vector lt that corresponds to a payment received of $\$1$ in period t and a payment made of $\$(1 + r)$ in period t + 1, with all other payments zero. Here r > 0 is the interest rate (over one period). Let c be a $\$1$ T − 1 period loan, starting at period 1. This means that $\$1$ is received in period 1, $(1 + r)^(T-1) is paid in period T, and all other payments (i.e., c2, . . . , cT −1) are zero. Express c as a linear combination of single period loans.

Answer 1

It is a single vector $c$ comprising entries as: $1, .., (1 + r)^{T −1}$
so, need to express $c$ as a linear combination of all these terms:
the linear combination ideally should have otherwise i.e. two vectors (one row vector or transpose of column vector) and second row vector.
Say, there are two vectors $s, w$ where $c = s^Tw$, and $s$ is a column vector of $1$ while, $w$ is a column vector of all the entries for correct multipliers. Also, the no. of terms in both $s,w$ are limited by $T$ as stated in the problem.