Calculate $tr(T)$ where $T(x,y,z)=(3x-z,2x+4y+2z,-x+3z)$

Question

Calculate $tr(T)$ where $T(x,y,z)=(3x-z,2x+4y+2z,-x+3z)$

I don't know how to calculate the trace of a linear operator.

I know $tr(A)=a_{11}+...+a_{nn}$

Let $B=\{e_1,e_2,e_3\}$ a basis of $\mathbb{R}^3$ then $T_{BB}=\begin{bmatrix} 3 &2 &-1 \\ 0&4 &0 \\ -1&2 &3 \end{bmatrix}$

My question:

Is $tr(T_{BB})=tr(T)$ true? If not, how do I calculate the trace of a linear operator?

Answer 1

you can use standard vector to find trace, $$T(1,0,0)=(3,2,-1)$$ $$T(0,1,0)=(0,4,0)$$ $$T(0,0,1)=(-1,2,3)$$ now trace of T is 3+4+3=10

Answer 2

Yes it is correct since trace is an invariant

$$T_{BB}=\begin{bmatrix} 3 &2 &-1 \\ 0&4 &0 \\ -1&2 &3 \end{bmatrix}\implies Tr(T)=10$$