"Let T P2→R3 be a transformation given by..." was answered by our community

High School
Geometry
High school geometry
Performing transformations

Maiclubk

Answered question

2021-03-02

Let T

P_{2} \to R^{3}

be a transformation given by

T (f (x)) = [\begin{matrix} f (0) \\ f (1) \\ 2 f (1) \end{matrix}]

(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.

Answer & Explanation

Elberte

Skilled2021-03-03Added 95 answers

The map T : $P_{2} \to R^{3}$ be a transformation given by
$T (f (x)) = [\begin{matrix} f (0) \\ f (1) \\ 2 f (1) \end{matrix}]$ .
(a)Let f, $g \in P_{2}$ and k is a real number. Then
$T (f (x) + k g (x)) = T (f (f + k g) (x))$
$= [\begin{matrix} (f + k g) (0) \\ (f + k g) (1) \\ 2 (f + k g) f (1) \end{matrix}]$
$= [\begin{matrix} f (0) + k g (0) \\ (f (1) + k g (1)) \\ 2 f (1) + 2 k g (1) \end{matrix}]$
$= [\begin{matrix} f (0) \\ f (1) \\ 2 f (1) \end{matrix}] + [\begin{matrix} k g (0) \\ k g (1) \\ 2 k g (1) \end{matrix}]$
$= T (f (x)) + k T (g (x)) .$
Therefore T: $P_{2} \to R^{3}$ is a linear transformation.
(b)
$K e r (T) = {f \in P_{2} : T (f (x)) = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]}$
$= {f \in P_{2} : [\begin{matrix} f (0) \\ f (1) \\ 2 f (1) \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]}$
$= {f \in P_{2} : f (0) = 0, f (1) = 0, 2 f (1) = 0}$
$= {f \in P_{2} : f (0) = 0, f (1) = 0}$
$= {f (x) = a x (x - 1), a \in R}$
Therefore Nullity $(T) = 1.$
Again Range(T)

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