# 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...

## 1 Answer

There was asked for only one matrix, namely the first one you got.
Note that writing out the coordinates of a vector in $$\Bbb R^3$$ implicitly uses the standard basis.
So, we take one basis in the domain (the polynomials), and one in the codomain (the real triples), and ask for the matrix of the transformation with respect to these bases.