Having some trouble with matrix norm 1

Question

So, I stumbled onto this exercise:

$T:(R^3, ||\cdot||_1)->(R^3, ||\cdot||_1)$

$T(x)=(2x_1-3x_2+x_3, x_1-x_3,2x_1+8x_2) for \ each x_i=(x_1,x_2,x_3) \in R^3$

Find ||T||.

I saw this as the matrix norm 1 and did $T_A(x):=Ax$ and tried to find $||A||_1$ which equals $||T_A|| $ but I don't really know if it's the right way to go about this because you can't get a proper A matrix from it.

Answer 1

I'm not sure what you mean by "you can't get a proper matrix $A$ from it". Note that $T(x) = Ax$ where $A$ is the matrix $$ A = \pmatrix{2 & -3 & 1\\ 1 & 0 & -1\\ 2 & 8 & 0} $$ Now, compute $\|A\|_1$ by the formula given here. You should find that $\|A\|_1 = 11$.