Hi the following assigment.(adsbygoogle = window.adsbygoogle || []).push({});

Given [tex]P_{2} (D)[/tex] be a vector space polynomials of at most degree n=2.

Looking at the transformation [tex]T: P_2(D) \rightarrow D^2[/tex], where

T(p) = [p(-i),p(i)].

1) Show that this transformation is linear.

I order to show this I hold my transformation up against the definition of the generel linear transformation?

2) The base [tex]B = (1,x,x^2)[/tex] in [tex]P_{2} (D) [/tex], and the standard base [tex]B' = (e_1,e_2)[/tex] in [tex]D^2[/tex]. Find the matrix representation for T in relation to the bases.

Since T(p) = [p(-i), p(i)], then there most be two polynomials of degree 2 which represent T:

[tex]T(p(-i),p(i)) = \[ \left( \begin{array}{c} 1 + i + i^2 & \\ 1 +(-i) + (-i)^2 & \\ \end{array}[/tex]

right?

Then the standard matrix representation for the transformation must be the matrix:

[tex] A(p(i)) = \[ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 1 & 0 & \\ 0 & 0 & 0 & 1 & 0 & \\ 0 & 1 & 0 & 0 & 0 & \\ 0 & 1 & 0 & 0 & 0 & \\ 1 & 1 & 3 & 0 & 0 & \\ \end{array} \right] \] [/tex]

but what about p(-i) then?

/Fred

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# Question regarding linear transformations

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