No idea where to start for linear transformation question from P2->P2

[jra0718]

Let T:P₂$\rightarrow$P₂ be given by

T(x-1)=1-x
T(x²-2x)=-1+x-2x²
T(3-x²)=-1+2x+3x²

Find the matrix for T with respect to the standard basis B={1,x,x²}for P₂

To be honest, I have no idea where to start. Help would be greatly appreciated

 

[ehild]

Write the polynomials as column vectors like
x-1 --->
-1,
1
0
x^2-2x --->
0
-2
1

and so on and make a matrix A from the three columns on the left hand side of the equation and a matrix B from the columns at the right-hand side. You get a matrix equation TA=B which you can solve for T
by multiplying both sides with the inverse of A:
T=TA A^-1=BA^-1.

ehild

 

[jra0718]

i don't understand how you got the column vectors

 

[ehild]

1, x, x^2 are the basis, consider them as base vectors e1, e2, e3. The coefficient of x^k in the polynomial is considered as the k-th component.

For example, -1+x-2x^2 = -1 (1) +1 (x) -2 (x^2) or -1 e1 +1*e2 -2 e3. You can write it as the vector (-1,1,-2), or its transposed as column vector.

ehild