Maiclubk

2021-03-02

Let T ${P}_{2}\to {\mathbb{R}}^{3}$ be a transformation given by
$T\left(f\left(x\right)\right)=\left[\begin{array}{c}f\left(0\right)\\ f\left(1\right)\\ 2f\left(1\right)\end{array}\right]$
(a)Then show that T is a linear transformation.
(b)Find and describe the kernel(null space) of T i.e Ker(T) and range of T.
(c)Show that T is one-to-one.

Elberte

The map T : ${P}_{2}\to {\mathbb{R}}^{3}$ be a transformation given by
$T\left(f\left(x\right)\right)=\left[\begin{array}{c}f\left(0\right)\\ f\left(1\right)\\ 2f\left(1\right)\end{array}\right]$.
(a)Let f,$g\in {P}_{2}$ and k is a real number. Then
$T\left(f\left(x\right)+kg\left(x\right)\right)=T\left(f\left(f+kg\right)\left(x\right)\right)$
$=\left[\begin{array}{c}\left(f+kg\right)\left(0\right)\\ \left(f+kg\right)\left(1\right)\\ 2\left(f+kg\right)f\left(1\right)\end{array}\right]$
$=\left[\begin{array}{c}f\left(0\right)+kg\left(0\right)\\ \left(f\left(1\right)+kg\left(1\right)\right)\\ 2f\left(1\right)+2kg\left(1\right)\end{array}\right]$
$=\left[\begin{array}{c}f\left(0\right)\\ f\left(1\right)\\ 2f\left(1\right)\end{array}\right]+\left[\begin{array}{c}kg\left(0\right)\\ kg\left(1\right)\\ 2kg\left(1\right)\end{array}\right]$
$=T\left(f\left(x\right)\right)+kT\left(g\left(x\right)\right).$
Therefore T:${P}_{2}\to {\mathbb{R}}^{3}$ is a linear transformation.
(b)
$Ker\left(T\right)=\left\{f\in {P}_{2}:T\left(f\left(x\right)\right)=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\right\}$
$=\left\{f\in {P}_{2}:\left[\begin{array}{c}f\left(0\right)\\ f\left(1\right)\\ 2f\left(1\right)\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\right\}$
$=\left\{f\in {P}_{2}:f\left(0\right)=0,f\left(1\right)=0,2f\left(1\right)=0\right\}$
$=\left\{f\in {P}_{2}:f\left(0\right)=0,f\left(1\right)=0\right\}$
$=\left\{f\left(x\right)=ax\left(x-1\right),a\in \mathbb{R}\right\}$
Therefore Nullity $\left(T\right)=1.$
Again Range(T)

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