Define a linear transformation by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 0 \right)}\end{array}} \right)\). Find \(T:{{\mathop{\rm P}\nolimits} _2} \to {\mathbb{R}^2}\)polynomials \({{\mathop{\rm p}\nolimits} _1}\) and \({{\mathop{\rm p}\nolimits} _2}\) in \({{\mathop{\rm P}\nolimits} _2}\) that span the kernel of T, and describe the range of T.

The kernel of \(T\) will contain any quadratic polynomial \({\mathop{\rm q}\nolimits} \) with \({\mathop{\rm q}\nolimits} \left( 0 \right) = 0\). The polynomial \({\mathop{\rm q}\nolimits} \) is \({\mathop{\rm q}\nolimits} = at + b{t^2}\).

Therefore, the polynomials \({{\mathop{\rm p}\nolimits} _1}\left( t \right) = t\) and \({{\mathop{\rm p}\nolimits} _2}\left( t \right) = t\) span the kernel of T.

When a vector is in the range of \(T\), it must be of the form \(\left( {\begin{array}{*{20}{c}}a\\a\end{array}} \right)\). When a vector is of the form, it is the image of the polynomial \({\mathop{\rm p}\nolimits} \left( t \right) = a\) in \({{\mathop{\rm P}\nolimits} _2}\).

Therefore, the range of \(T\) is \(\left\{ {\left( {\begin{array}{*{20}{c}}a\\a\end{array}} \right):a\,\,{\mathop{\rm real}\nolimits} } \right\}\).