Description
Instructions: Read textbook pages 57 to 59 before working on the homework problems. Show all steps to get full credits.
1. Let f : P
3 → R be a mapping with f(a0 + a1x + a2x
2 + a3x
3
) = a3 for all
a0 + a1x + a2x
2 + a3x
3
in P
3
. Prove that f is a linear mapping.
2. For each of the following matrices
A =
Ñ
0 0 0
0 2 0
0 0 −8
é
, B =
Ñ
−1 0 0
3 2 0
4 5 3é
, C =
Ñ√
2 i 1 − 2i
0 2 − 3i 2 + i
0 0 1 é
,
D =
Ñ
1 2 3
2 4 −i
3 −i 0
é
, E =
Ñ
1 1 + i 2 − i
1 − i 2 4
2 + i 4 3 é
, F =
Ñ
1 −2 3 4
0 −2 3 5
0 0 0 0é
,
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.
