Hi guys, can anyone help me with this exercise. My exam is coming soon and i need this exercise completed for compare with my results and feel sure :S
It says:
1)We define = f: P₂[R] ---> R^2x2 linear transformation whose transformation matrix in basis B and E' is:
Mf(over BE')=
[1...2..-1]
[1...0...1]
[4...8..-4]
[2...2...0]

Where B= {1, -X, 2X^2 + 1} and E', the canonical basis of R^2x2.
a)Find subspaces Ker(f) and Im(f).
b)Using the matrix of change of coordinates, find Mf(over EE') - where E = {X^2, X, 1}.

2)The linear transformation T:R^3 ---> R^4, satisfies the following conditions:
T((1,0,0)) = (0,0,0,0).
T((0,2,-1)) = T((3,0,2)) = (0,1,1,0).
a)If possible, find a formula for T(v) for all v ∈ R^3.
b)Identify a pair of bases of vector spaces domain and codomain respectively, and find the matrix of T with respect to those chosen bases.

Thanks!