Find polynomial p1andp2∈P2 that spans the kernel of T and describes the range of sf "T".

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fortdefruitI

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2020-11-20

Find polynomial $p_{1} and p_{2} \in P_{2}$ that spans the kernel of T and describes the range of sf "T".

Answer & Explanation

Sadie Eaton

Skilled2020-11-21Added 104 answers

The linear transformation T is defined as sf $T : P_{2} \to R^{2} b y (p) = [\begin{matrix} p & (0) \\ p & (0) \end{matrix}]$
The set of polynomial p in $P^{2} such that T (p) 0,$ are the members of kernel of T.
Therefore, any quadratic polynomial p with properties $p (0) = 0$ will be in kernel of T.
Thus, the two polynomials $p_{1} (t) = t and p_{2} (t) =$ wil be in kernel of T.
The polynomial t and $t^{2}$ linearly independent.
Thus, the linear combination of t and $t^{2}$ will be in kernel of T.
Hence, the polynomials $p_{1} (t) = t and p_{2} (t) = t^{2}$ spans the kernel of T.
The set of all polynomial contained in range of T is ${p \in V : T ((p)}$
The value of p (0) is always constant.
Thus, the range of T is given by ${[\begin{matrix} c \\ c \end{matrix}] : c i s \in R} .$
The range of T is ${[\begin{matrix} c \\ c \end{matrix}] : c i s \in R} .$

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