Instructions: Read textbook pages 57 to 59 before working on the homework problems. Show all steps to get full credits.
- 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
| 0 0A = 0 20 0 | 0 0 ,B =8 | 1 0 0 2 i3 2 0 ,C = 0 2 3i4 5 3 0 0 | 1 2i2 + i ,1 |
| 1 2D = 2 43 i | 3 i ,E =0 | 1 1 + i 2 i 11 i 2 4 ,F = 02 + i 4 3 0 | 2 3 42 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 , some coefficients, we have (A + B)T = AT + BT. 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.
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