Linear algebra proof

[buckaroobill]

If anyone could explain how the following proof on linear transformations/isomorphism is done, it would be greatly appreciated!

Show that L: P_n -> P_n given by L(p) = p + p' is an isomorphism. First show that L is a linear operator.

 

[daon]

If anyone could explain how the following proof on linear transformations/isomorphism is done, it would be greatly appreciated!

Show that L: P_n -> P_n given by L(p) = p + p' is an isomorphism. First show that L is a linear operator.

Your "isomorphism" depends on p' and it will matter what it means. For example if p' = -p then you have a trivial homomorphism.

 

[pka]

It is likely that \(\displaystyle P_n\) is the space of polynomials of degree \(\displaystyle \le n\) and \(\displaystyle p'\) is the derivative. Proving that \(\displaystyle L\) is linear is the easy part, just note that the derivative operator is linear. It is more difficult to show the other properties of an isomorphism. Basically, show that the kernel is \(\displaystyle \left\{ 0 \right\}\). To do that note that \(\displaystyle \left\{ {1,x,x^2 , \cdots ,x^n } \right\}\) is a linearly independent set.