Let $w_1 = [1, 2, 0]$, $w_2 = [2, 5, 1]$, $w_3 = [2, 4, 1]$ Let $\def\R{\Bbb R}f : \R^3\to \R^3$ be the linear transformation satisfying $f(w_1) = w_2-w_3$, $f(w_2) = -w_2+w_3$, $f(w_3) = w_1+w_2+w_3$
a) Give the matrix representation of $f$ with respect to the basis $\{w_1, w_2, w_3\}$
b) Give the matrix representation of $f$ where the input $x$, is written with respect to the basis $\{w_1, w_2, w_3\}$ and the output $f(x)$ is written with respect to the basis $\{e_1, e_2, e_3\}$ (the standard basis)
c) Is $w_1$ in the range of $f$?
Attempt:
a) I think for this part I just have to do Gaussian elimination of the matrix $[w_1][w_2][w_3]$ correct ?
b/c) not sure how to do them
For a), the columns of your matrix should express the images $f(w_1)$, $f(w_2)$ and $f(w_3)$ in coordinates with respect to the basis $\{w_1,w_2,w_3\}$. Since those images are already given expressed in terms of $w_1,w_2,w_3$, finding the coordinates of an image just amount to picking up the coefficients of $w_1$, $w_2$, and $w_3$ respectively from the given expression.
For b) one asks to use the standard basis at arrival. This means you need to rewrite the given images in terms of $e_1,e_2,e_3$ (using the given expressions for the $w_i$) and then again pick up respective coefficients to fill your matrix.
For c) you need to see if $f(xw_1+yw_2+zw_3)=w_1$ has any solutions for $x,y,z\in\R$. You can do this by writing the linear system whose left hand side is the matrix from a), and whose right hand side is the column $(1,0,0)$ (the expression for $w_1$ on the basis $\{w_1,w_2,w_3\}$) and doing Gaussian elimination.