I have dosed off in class way too many times and I need help with a few problems from my pre-test:
#3
Prove that M_22 is isomorphic to P_3.
A simple hint in the right direction would be greatly appreciated.
Advathanksnce!
This isn't what you are trying to do. You want to find the kernel of the map L, which will be a 4-by-4 matrix but you don't want to think of it that way. You need to solve,
$\displaystyle L(M)=0 \Rightarrow \ldots$.
This is basically asking you if $\displaystyle ax^3+bx^2+cx+d = 0$ then what are $\displaystyle a, b, c$ and $\displaystyle d$?
No, 4-by-4. You are mapping from a 4-dimensional vector space into a 4-dimensional vector space...
The matrix you are mapping from is 2-by-2, and I was presuming the OP was getting confused by the linear map ~ matrix equivalence (it seemed like he was trying to find the kernel of the 2-by-2 matrix). That is why I wanted to point out that T is 4-by-4.