Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 32E from Chapter 4.2 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 32E
Chapter:
Problem:
Define a linear transformation Find polynomials p1 and p2 in that span the kernel of T , and describe the range of T .
Step-by-Step Solution
Given Information
We are given with a transformation \[T:{P_2} \to {R^2}::\,\,T({\bf{p}}) = \left[ {\begin{array}{*{20}{l}} {{\bf{p}}(0)}\\ {{\bf{p}}(0)} \end{array}} \right]\] We also have to find a polynomial that spans kernel of T.
Step-1: (a)
Let us assume the polynomial $p$ is. \[p(t) = a + bt + c{t^2}\] Apply the conditions to polynomial. \[ \begin{aligned} \mathbf { p } ( 0 ) & = a + b ( 0 ) + c ( 0 ) ^ { 2 } \\ & = a + 0 + 0 \\ \mathbf { p } ( 0 ) & = a \end{aligned} \] Hence, all polynomials of the form $bt+ct^2$ span the kernel of transformation.
Hence kernel of the transformation is $p(t) = bt + c{t^2}$
We are given with a transformation \[T:{P_2} \to {R^2}::\,\,T({\bf{p}}) = \left[ {\begin{array}{*{20}{l}} {{\bf{p}}(0)}\\ {{\bf{p}}(0)} \end{array}} \right]\] We also have to find a polynomial that spans kernel of T.
Step-1: (a)
Let us assume the polynomial $p$ is. \[p(t) = a + bt + c{t^2}\] Apply the conditions to polynomial. \[ \begin{aligned} \mathbf { p } ( 0 ) & = a + b ( 0 ) + c ( 0 ) ^ { 2 } \\ & = a + 0 + 0 \\ \mathbf { p } ( 0 ) & = a \end{aligned} \] Hence, all polynomials of the form $bt+ct^2$ span the kernel of transformation.
Hence kernel of the transformation is $p(t) = bt + c{t^2}$