MATH307 Individual Homework11 Solved

Description

Instructions: Read textbook pages 57 to 59 before working on the homework problems. Show all steps to get full credits.

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 = A^T. 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 = αA^T + βB^T. 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.