Let $V$ be a vector space, and $$T:V→V$$ a linear transformation such that $$T(2v_1−3v_2)=−3v_ 1+2v_ 2$$ and $$T(−3v_ 1+5v_2)=5v_ 1+4v_2$$
Solve $$T(v_1), T(v_2), T(−4v_ 1−2v_2)$$
I tried to put it into an Augmented matrix, take the inverse, and apply the inverse in order to work backwards from there but I still don't have any luck.. Does anyone have tips?
Best Answer
You list some very overpowered things that you don't need for this problem. Remember that linearity is a very strong condition.
Hint: how do you "make" $v_1$ out of copies (linear combinations) of $2v_1-3v_2$ and $-3v_1+5v_2$?
Now, can you do the other problems?