# Thread: Linear Transformation in polynomial vector space

1. ## Linear Transformation in polynomial vector space

Define T: P1 --> P1 by

T(ax + b) = (2b - a)x + (b + a)

Show that T is both one-to-one and onto, and find the inverse transformation to T.

This problem showed up in my homework, and I'm not sure how to finish it. I was able to prove that T is both on-to-one and onto, but don't know how to find T^(-1). I hope someone can help. Thanks in advance!

2. Originally Posted by dan213
Define T: P1 --> P1 by

T(ax + b) = (2b - a)x + (b + a)

Show that T is both one-to-one and onto, and find the inverse transformation to T.

This problem showed up in my homework, and I'm not sure how to finish it. I was able to prove that T is both on-to-one and onto, but don't know how to find T^(-1). I hope someone can help. Thanks in advance!
Let $\displaystyle e_1=1$ and $\displaystyle e_2=x$ be the basis vectors for
$\displaystyle \mathbb{P}_1$

Then $\displaystyle T(1)=T(e_1)=2x+1=e_1+2e_2$
and
$\displaystyle T(x)=T(e_2)=-x+1=e_1-e_2$

So the matrix representation of the linear transformation is

$\displaystyle \begin{bmatrix}1 & 1 \\ 2 & -1 \end{bmatrix}$

Can you finish finding the inverse from here?

3. Yes I can - thank you. I appreciate your help.