# Transforming matrix for a linear transformation:

Linear transformation: $$T:\mathbb{R}[X]_{≤2}\rightarrow\mathbb{R}^3$$

With $$T(f):=(f(0), f'(1),f(2))$$ and $$\mathbb{R}[X]_{≤2}$$ is the $$\mathbb{R}$$-vector space of the polynomials of degree ≤2

Portray the linear transformation $$T$$ as matrix referring to the basis $$1, X, X^2$$ of $$\mathbb{R}[X]_{≤2}$$ and the standard basis of $$\mathbb{R}^3$$

Now I've done the transformation for both the bases, but I don't know if that works:

For the first one I obtain a matrix:

$$[1,0,0]$$

$$[0,1,2]$$

$$[1,2,4]$$

Doing $$T(1)=(1,0,1)$$, $$T(X)=(0,1,2)$$, $$T(X^2)=(0,2,4)$$

And for the standard basis I obtained a matrix:

$$[1,0,0]$$

$$[0,0,0]$$

$$[0,0,1]$$

Doing the same procedure with $$T(e_1), T(e_2), T(e_3)$$

I'm not sure it's right, and I don't know if I have to get just one another matrix instead of one. Sorry I don't know how to write matrices here...

Note that writing out the coordinates of a vector in $$\Bbb R^3$$ implicitly uses the standard basis.