Description
- 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.
- 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.
- 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.
- Prove that diagonal entries of Hermitian matrices have to be real valued.
