Let $V=R^3$ and $T:V\rightarrow V$ be the linear map defined by
$T(x,y,z)=(x+z, -2x+y, -x+2y+z)$.
What is the matrix of $T$ wrt the basis $(1,0,1)$, $(-1,1,1)$ and $(0,1,1)$.
Using this matrix, write down the matrix of $T$ wrt the basis $(0,1,2)$, $(-1,1,1)$ and $(0,1,1)$.
I got the first part.
For the second part, how do I use the matrix of $T$ wrt one basis to arrive at the matrix of $T$ wrt another basis?
Please help.
Let $B=\{b_1,b_2,b_3\}$ and $B'=\{b'_1,b'_2,b'_3\}$ be bases. Let $P$ be the matrix of the linear map $g$ with $g(b_i)=b'_i$ for $i=1,2,3.$ When $f$ is a linear map, and $T$ is the matrix of $f$ with respect to $B$, the matrix $P^{-1}TP$ is the matrix of $f$ with respect to $B'$.
Because if a vector $v$ is represented by a vertical co-ordinate sequence $S$ with respect to $B'$, then $PS$ calculates the co-ordinates of $v$ with respect to $B.$ Then $T(PS)=TPS$ calculates the co-ordinates of $f(v)$ in basis $B.$ And $P^{-1}(TPS)$ calculates what the co-ordinates of $f(v)$ are in basis $B'$.
Of course this applies in any number of dimensions.
In your Q , $B=\{(1,0,0),(0,1,0), (0,0,1)\}$ and $B'=\{(1,0,1),(-1,1,1),(0,1,1)\}.$
What is confusing in this subject is that we often speak of $(a,b,c)$ as a vector but also refer to $(a,b,c)$ as the representation of a vector with respect to some basis. It would be better if we put an asterisk on one of these types of things to distinguish it from the other, but nobody does. The same can be said for maps and matrices.