Throughout this question B = will be the basis for given by:
and S will denote the standard basis
If T : R3 R3 is the linear map which is determined by:
T(v1) = v3 , T(v2) = v2 - v1 and T(v3) = -v3 ,
write down the matrix representation [T]BB
Throughout this question B = will be the basis for given by:
and S will denote the standard basis
If T : R3 R3 is the linear map which is determined by:
T(v1) = v3 , T(v2) = v2 - v1 and T(v3) = -v3 ,
write down the matrix representation [T]BB
Well, the most direct way would be
$\displaystyle Tv_1 = \left [ \begin{matrix} a & b & c \\ d & e & f \\ g & h & k \end{matrix} \right ] \left [ \begin{matrix} 1 \\ -1 \\ 0 \end{matrix} \right ] = \left [ \begin{matrix} 0 \\ 1 \\ 1 \end{matrix} \right ] $
This implies the system
$\displaystyle a - b = 0$
$\displaystyle d - e = 1$
$\displaystyle g - h = 1$
The other two relations give you the other six equations you need to solve for T.
-Dan
$\displaystyle Tv_1 \neq v_3$
Sorry I don't have better news.
You already have the $\displaystyle Tv_1 = v_3$ conditions. I also get
$\displaystyle a + 2c = 0$
$\displaystyle d + 2f = 1$
$\displaystyle g + 2k = 2$
$\displaystyle b + c = 0$
$\displaystyle e + f = -1$
$\displaystyle h + k = -1$
The first row is right, though.
-Dan