How does one write an transformation matrix under an orthonormal basis?

Question

I have an orthonormal basis (consisting of either vectors or polynomials). How do I write a matrix for a linear transformation $Tv=w$ for any vector $v$ and $w$ in the vector space $V$? Is there a general process for doing this?

For example, $T(x_1,x_2,x_3)=(3x_1,2x_3,x_2)$ under orthonormal basis matrix \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}

would be written as:

\begin{pmatrix}3 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 2 & 0\end{pmatrix}

Is that correct?

Answer 1

just apply the transformation to the basis vectors, express the images as column vectors, and put them together to form the matrix. so the correct matrix should be $$ \left( \begin{array}{ccc} 3 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 2 & 0 \end{array} \right) $$

Answer 2

For any basis, the columns of the transformation matrix are the images—expressed relative to that basis—of the basis vectors.