This is not really a homework but I am preparing for a midterm and I am stuck on a problem from the book. I have the solution but I can't understand it.
I have to computer $[T]_\beta^\gamma$.
This is the solution :
Now I understand where $T(1,0) = (1,1,2)$ Comes from.
The problem is the part after. $-1/3 (1,1,0)$ . I get the $ 1,1,0$ comes from the $\gamma$ symbol. But where does the $1/3$ comes from? Same thing for the $2/3$ and the other numbers infront.
Any help would be appreciated.
Consider the equation$$T(1,0)=a(1,1,0)+b(0,1,1)+c(2,2,3).$$It can be written as$$\left\{\begin{array}{l}a+2c=1\\a+b+2c=1\\b+3c=2.\end{array}\right.$$If you solve it, you'll get $a=-\dfrac13$, $b=0$, and $c=\dfrac23$. That's where those numbers come from.