This expression leaves us however with some terms with negative indices, for example \( x(-1) \) and \( x(-2) \) which may not be defined. Our vector \( \boldsymbol{x} \) has components \( x(0) \), \( x(1) \), \( x(2) \) and \( x(3) \).
The index \( j \) for \( \boldsymbol{x} \) runs from \( j=0 \) to \( j=3 \) since \( \boldsymbol{x} \) is meant to represent a third-order polynomial.
Furthermore, the index \( i \) runs from \( i=0 \) to \( i=5 \) since \( \boldsymbol{y} \) contains the coefficients of a fifth-order polynomial. When \( i=5 \) we may also have values of \( x(4) \) and \( x(5) \) which are not defined.