Huge linear system of equations with powers of $2$:

$$\begin{array}{rl}({2}^{0}{)}^{n}{a}_{n}+({2}^{0}{)}^{n-1}{a}_{n-1}+\cdots +({2}^{0}{)}^{1}{a}_{1}& ={4}^{0}\\ ({2}^{1}{)}^{n}{a}_{n}+({2}^{1}{)}^{n-1}{a}_{n-1}+\cdots +({2}^{1}{)}^{1}{a}_{1}& ={4}^{1}\\ \vdots \\ ({2}^{n-1}{)}^{n}{a}_{n}+({2}^{n-1}{)}^{n-1}{a}_{n-1}+\cdots +({2}^{n-1}{)}^{1}{a}_{1}& ={4}^{n-1}\end{array}$$

for $n\ge 2$. Show that the (unique) solution of this system is when ${a}_{2}=1$ and all other variables are zero.

$$\begin{array}{rl}({2}^{0}{)}^{n}{a}_{n}+({2}^{0}{)}^{n-1}{a}_{n-1}+\cdots +({2}^{0}{)}^{1}{a}_{1}& ={4}^{0}\\ ({2}^{1}{)}^{n}{a}_{n}+({2}^{1}{)}^{n-1}{a}_{n-1}+\cdots +({2}^{1}{)}^{1}{a}_{1}& ={4}^{1}\\ \vdots \\ ({2}^{n-1}{)}^{n}{a}_{n}+({2}^{n-1}{)}^{n-1}{a}_{n-1}+\cdots +({2}^{n-1}{)}^{1}{a}_{1}& ={4}^{n-1}\end{array}$$

for $n\ge 2$. Show that the (unique) solution of this system is when ${a}_{2}=1$ and all other variables are zero.