The linear transformation $g: R^3 \rightarrow R^3 $ given by $$g(x,y,z) = (x+y,~2y+z,~3z)$$ Find a basis $T$ for $R^3$ such that $_T[f]_T$ is diagonal.
First I found a matrix of a linear transformation
$$_S[g]_S= \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \end{pmatrix} $$
and $g$ has eigenvalues $1,2,3$.
But know I am not sure how to find a basis $T$.
Should I try to find the eigenspaces $V(\lambda)$-s corresponding to eigenvalues?
I'm not really sure how, to begin with, this one. Anything would help thanks.