<s>[INST] QR factorization for least squares This is from my textbook

I don't undertand why small errorr in $A^TA$ can lead to large error in cofficient matrix? Because A=QR, so there should be no difference to use A or QR anyway.Could someone give an example? Thank you very much
 [/INST] A typical example uses the matrix
$$
\mathbf{A} =
\left(
\begin{array}{cc}
 1 & 1 \\
 0 & \epsilon  \\
\end{array}
\right)
$$
Consider the linear system
$$
\mathbf{A} x= b.
$$
The solution via normal equations is
$$
%
\begin{align}
%
 x_{LS} &= \left( \mathbf{A}^{*}\mathbf{A} \right)^{-1}\mathbf{A}^{*}b \\
% x
\left(
\begin{array}{cc}
 x_{1} \\
 x_{2} 
\end{array}
\right)_{LS}
%
&=
%
%
\left(
\begin{array}{cc}
 b_{1}-\frac{b_{2}}{\epsilon }\\\frac{b_{2}}{\epsilon } 
\end{array}
\right)
%
\end{align}
%
$$
Minute changes in $\epsilon$, for example, $0.001\to0.00001$ create large changes in the solution:
$$
  \epsilon = 0.001: \quad x_{LS} =
\left(
\begin{array}{cc}
 b_{1}-1000b_{2} \\ 1000 b_{2}
\end{array}
\right)
$$
$$
  \epsilon = 0.00001: \quad x_{LS} = 
\left(
\begin{array}{cc}
 b_{1}-100000b_{2} \\ 100000 b_{2}
\end{array}
\right)
$$
The $\mathbf{QR}$ decomposition is
$$
 \mathbf{A} = \mathbf{QR} =
\left(
\begin{array}{cc}
 1 & 0 \\
 0 & \frac{\epsilon }{\left| \epsilon \right| } \\
\end{array}
\right)
%
\left(
\begin{array}{cc}
 1 & 1 \\
 0 & \left| \epsilon \right|  \\
\end{array}
\right)
$$
</s>