1. Let f : P3 →R be a mapping with f(a0 + a1x + a2x2 + a3x3) = a3 for all a0 + a1x + a2x2 + a3x3 in P3. Prove that f is a linear mapping.
2. For each of the following matrices
Ñ0 0
A = 0 2
0 0
0 é
0 ,B =
−8
√
Ñ−1 0 0é Ñ 2 i
3 2 0 ,C = 0 2 − 3i
4 5 3 0 0
1 − 2ié
2 + i ,
1
Ñ1 2
D = 2 4
3 −i
3 é
−i ,E =
0
Ñ 1 1 + i 2 − ié Ñ1
1 − i 2 4 ,F = 0
2 + i 4 3 0
−2 3 4é
−2 3 5 ,
0 0 0
specify whether it is diagonal, upper-triangular, lower-triangular, symmetric or hermitian. Note one matrix might have more than one structures. For instance, a diagonal matrix is also upper-triangular. Moreover, a matrix is symmetric if A = AT. It applies to complex matrices as well.
3. Prove that for two matrices A,B of the same size and α,β some coefficients, we have (αA + βB)T = αAT + βBT. Note, to prove two matrices are equal, it suffices to prove the ij-th entry of the two matrices are equal for all legal indices i,j.
4. Prove that diagonal entries of Hermitian matrices have to be real valued.
