Determine whether a Linear Transformation is Invertible
Hey everyone, I have a final exam tomorrow and I would greatly appreciate it if someone could help me out with this question:
For each of the following linear transformations T, determine whether T is invertible and justify your answer.
1. T:R^2 -> R^3 defined by T(a_1, a_2) = (a_1 - 2a_2, a_2, 3a_1 + 4a_2).
2. T:R^3 -> R^3 defined by T(a_1, a_2) = (3a_1 - a_2, a_2, 4a_1).
3. T:M_2x2(R) -> M_2x2(R) defined by T(a b c d) = (a+b a c c+d).
Both of these are 2x2 square matrices so "a" corresponds to the (1,1) entry, "b" corresponds to the (1,2) entry, "c" corresponds to the (2,1) entry, and "d" corresponds to the (2,2) entry. Same with "a+b", "a", etc.
I think these deal with isomorphisms but I'm not sure. Also, there is a theorem that says T is invertible iff T is one-to-one and onto. But how do I prove that T is one-to-one and onto?
Thanks for you help. BTW, what happened to Latex?