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.

Short Answer

Expert verified

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

Step by step solution

Determine the polynomials ${{\mathop{\rm p}\nolimits} _1}$ and ${{\mathop{\rm p}\nolimits} _2}$ that span the kernel 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.

Describe the range 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 $\left\{ {\left( {\begin{array}{*{20}{c}}a\\a\end{array}} \right):a\,\,{\mathop{\rm real}\nolimits} } \right\}$.