hello can someone explain this question to me?
question:
if $g: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ satisfies $$[g]_u = \left[ \begin{array}{ccc} 1&0&0\\ 0&-1&0\\ 0&0&2 \end{array} \right] $$
compute $g(x_1, x_2, x_3)$ if $u=\{(1,1,1), (1,-1,1), (1,1,-1)\}$
There is more than one way of doing that. Assuming $(x_1,x_2,x_3)$ denotes an $\mathbb{R}^3$ vector with coordinates in the canonical basis, one way of doing this is to find the matrix of $g$ with respect to the canonical basis, which should be done by left-multiplying $[g]_u$ by the matrix whose columns are the vectors of $u$ with respect to the canonical basis (the columns should be in the right order, i.e, the first column should be $(1,1,1,)^T$, and so on), and by right-multiplying it by the inverse of the matrix that you left-multiplied. By doing this, you will get $[g]_c$, with $c$ being the canonical basis; so all is left is calculate $g(x_1,x_2,x_3)$, which you can do by right-multiplying $[g]_c$ by $(x_1,x_2,x_3)^T$.