# Thread: Linear Algebra of polynomials

1. ## Linear Algebra of polynomials

I need help with this specific problem.

a) prove that T is linear
b) give the kernel dimension of T et dertermine a base of the image of T
c) The application T is it injective? Is T surjective?

2. ## Re: Linear Algebra of polynomials

Originally Posted by steelmaste

I need help with this specific problem. :
a) prove that T is linear
b) give the kernel dimension of T et dertermine a base of the image of T
c) The application T is it injective? Is T surjective?
$\displaystyle T(a{x^2} + bx + c) = \left( {\begin{array}{*{20}{c}} {p(0)}&{p'(0)}\\ {p'(0)}&{p''(0)} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} c&b\\ b&{2a} \end{array}} \right)$
a) It should be easy to show $T(\alpha p+q)=\alpha T(p)+T(q)$

b) Looking at the mapping what polynomial(s) map to $\displaystyle \left( {\begin{array}{*{20}{c}} 0&0\\ 0&0 \end{array}} \right)~?$

c) for injectivity If T(p)=T(q) does it follow that $p=q$ HINT: what if $\displaystyle \left( {\begin{array}{*{20}{c}} c&b\\ b&2a \end{array}} \right)=\left( {\begin{array}{*{20}{c}} f&e\\ e&2d \end{array}} \right)~?$

For surjectivity what happens in the case of $\displaystyle \left( {\begin{array}{*{20}{c}} 0&2\\ 1&3 \end{array}} \right)~?$

3. ## Re: Linear Algebra of polynomials

$\displaystyle T(1),T(x),T\left(x^2\right)$ are linearly independent matrices ("vectors")

so $dim Im T =3$

therefore $dim Ker T = 0$

and $T$ is injective