# Change of basis help?

1. Dec 9, 2012

### bonfire09

1. The problem statement, all variables and given/known data
Problem is assuming the mapping T: P2---->P2 defined by T(a0+a1t+a2t2)=3a0+(5a0-2a1)t+(4a1+a2)t^2 is linear. Find the matrix representation of T relative to Basis B={1,t,t^2}.
The part that im confused on is when I go plug in the basis values T(1),T(t),and T(t^2)? I don't know how to do it?

2. Relevant equations

3. The attempt at a solution

So to find T(1) its just T(1+0t+0t2)=3a0+5a0t

To find T(t) is just T(0+a1(t)+0T2)=3(0)+(5(0)-2a1)t+(4a1+0)t^2=-2a1t+4a1t^2

T(t^2)= T(0+0t+a2t^2)=3(0)+(5(0)-2(0))t+(4(0)+a2)t^2=a2T2

Usually in lots of books they omit steps like these and I'm trying to figure them out. Is this a correct way?

2. Dec 9, 2012

### pasmith

This is the right idea, but to get $T(1)$ you take $a_0 = 1$, $a_1 = 0$, and $a_2 = 0$ so that $T(1) = 3 + 5t$. Similarly for the other two basis vectors.

3. Dec 9, 2012

### bonfire09

Oh ok. So for T(t) just let a0=0, a1=1 and a2=0 and for T(t^2) just let a0=0,a1=0 and a2=1?

That looks like the standard basis {e1,e2,e3}