Linear Mapping T: P2 to P2 with Basis B | Homework Help & Solution Explained"

[bonfire09]

Homework Statement

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 I am 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?

Homework Equations

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?

 

[pasmith]

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?

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.

 

[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}