Given the following linear transformation:
$$f : \mathbb{R}^2 \to \mathbb{R}^3 | f(1; 0) = (1; 1; 0), f(0; 1)=(0; 1; 1)$$
find the associated matrix of $f$ with respect of the following basis:
$R = ((1; 0); (0; 1))$ of $\mathbb{R}^2$
and
$R^1 =
((1; 0; 0); (1; 1; 0); (0; 1; 1))$ of $\mathbb{R}^3$
—
I've found the associated matrix of $f$ with respect of the basis $R$, and it appears to be the following:
$$
\begin{bmatrix}1&0\\1&1\\0&1\end{bmatrix}
$$
Is it correct? How can I calculate the associated matrix of $f$ with respect of the basis $R^1$?
Best Answer
Your matrix $$ \begin{bmatrix}1&0\\1&1\\0&1\end{bmatrix} $$ represent the function $f$ if we use the standard basis $R_3=((1,0,0);(0,1,0);(0,0,1))$ of $\mathbb{R}^3$. If you use the basis $R_3^1=((1,0,0);(1,1,0);(0,1,1))$ note that $(1,0)$ become the second element of this new basis, and $(0,1)$ become the third element, so the matrix become:
$$ \begin{bmatrix}0&0\\1&0\\0&1\end{bmatrix} $$