1. ## Linear map

Show that the linear map $T:\wp_2 \to \wp_2$ defined by $T(\alpha_0+\alpha_1 x+\alpha_2x^2)=(\alpha_0+\alpha_1)+(\alpha_1+2\alp ha_2)x+(\alpha_0+\alpha_1+\alpha_2)x^2$ is non-singular and find its inverse.

2. Originally Posted by kjchauhan
Show that the linear map $T:\wp_2 \to \wp_2$ defined by $T(\alpha_0+\alpha_1 x+\alpha_2x^2)=(\alpha_0+\alpha_1)+(\alpha_1+2\alp ha_2)x+(\alpha_0+\alpha_1+\alpha_2)x^2$ is non-singular and find its inverse.

It is hard to answer a question like that when we do not know your definition of singular linear transformation or the theorems you know.
Try one of these:
1)Show the determinant of T is non zero by computing the matrix of T (fixing a basis) and then invert the matrix and then finally get inverse of T.
2)Show that T is one-one and onto. Let F be the inverse map, then $T\circ F = \text{Id}$. So compute F directly from that.

The second method is messy but it only uses definition of inverse.