Math 1080: Homework #1 solved

Description

Problem 1:
Let 𝐵 be a 4×4 matrix to which we apply the following operations:
1. Double column 3
2. Add row 2 to row 1
3. Interchange columns 2 and 3
4. Halve row 4
5. Replace column 4 by sum of columns 1 and 3
Each of these operations can be performed by multiplying 𝐵 on the left or on the right by
a specific matrix 𝐸𝑘 (where 𝑘 stands for the operation above) Find the matrices 𝐸𝑘. Then
find matrices 𝐴 and 𝐶 such that the result is obtained as a product 𝐴𝐵𝐶
Problem 2:
Consider the matrix
𝑄 =
1
3
[
2 −1 2
2 2 −1
−1 2 2
]
Show that Q is an orthogonal matrix. What transformation of
3
IR
does it correspond to?
(Hint: Find the vector a that is invariant under Q. Pick a vector b orthogonal to a. Find
the angle α between b and Qb. If this angle is independent of the choice of b, then Q
corresponds to a rotation about a by the angle α. Think about other possibilities.)
Problem 3:
Find the 2×2 orthogonal matrix Q that corresponds to the reflection over the line
2𝑥 − 3𝑦 = 0.
Problem 4:
Let 𝑢, 𝑣 be two vectors and 𝐴 = 𝐼 + 𝑢𝑣
𝑇
a matrix. Show that if 𝐴 is invertible, its inverse
is the matrix 𝐴
−1 = 𝐼 + 𝛼𝑢𝑣
𝑇
and find the scalar 𝛼.When is 𝐴 singular?
Problem 5:
(a) Compute the norms ‖𝑤‖1, ‖𝑤‖2, ‖𝑤‖∞ for the vector 𝑤 = [
3
−1
5
]
(b) Compute the norms ‖𝐴‖1, ‖𝐴‖2, ‖𝐴‖∞ for the matrix 𝐴 = [
2 −1 1
−1 0 2
]
(c) Verify the inequalities ‖𝐴𝑤‖𝑝 ≤ ‖𝐴‖𝑝‖𝑤‖𝑝.