# Finding Inverse of a Transformation

1. Mar 11, 2012

### TranscendArcu

1. The problem statement, all variables and given/known data

3. The attempt at a solution
Suppose we take an arbitrary polynomial in $P_2 (R)$, call this $a_0 + a_1 x + a_2 x^2$

$T(a_0 + a_1 x + a_2 x^2) = (a_0, a_0 + a_1 + a_2, a_0 - a_1 + a_2)$

Now, I was under the impression that I could construct a matrix for T by showing what T does to each of the standard basis vectors of $P_2 (R)$, these being the set $1,x,x^2$.

Doing some mapping: $T(1 + 0x + 0x^2) = (1,1,1), T(0 + 1x + 0x^2) = (0,1,-1), T(0+0x+1x^2) = (0,1,1)$.

Thus my matrix for T has these outputs as its columns. Via some elementary row reduction I concluded that the matrix (with semicolons indicating the end of a row) [1 0 0;0 1/2 1/2;-1 1/2 -1/2] is the inverse of T, call it S.

Moreover, it can be checked that T*S = I, which leads me to suspect that I have the write answer. Does this look about right?

2. Mar 11, 2012

Yes it does