Homework #1 Solution

$30.00

Category: Tag:

Description

Problem 1:

Let be a 4×4 matrix to which we apply the following operations:

  1. Double column 3

  1. Add row 2 to row 1

  1. Interchange columns 2 and 3

  1. Halve row 4

  1. Replace column 4 by sum of columns 1 and 3

Each of these operations can be performed by multiplying on the left or on the right by a specific matrix (where stands for the operation above) Find the matrices . Then find matrices and such that the result is obtained as a product

Problem 2:

Consider the matrix

1

2

1

2

=

[ 2

2

1]

3

1

2

2

Show that Q is an orthogonal matrix. What transformation of IR3 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 2 − 3 = 0.

Problem 4:

Let , be two vectors and = + a matrix. Show that if is invertible, its inverse is the matrix −1 = + and find the scalar .When is singular?

Problem 5:

3

  1. Compute the norms ‖ ‖1, ‖ ‖2, ‖ ‖ for the vector = [−1] 5

(b) Compute the norms ‖ ‖ , ‖ ‖

2

, ‖ ‖

for the matrix = [ 2

1

1]

1

1

0

2

(c) Verify the inequalities ‖ ‖

≤ ‖ ‖ ‖ ‖ .


error: Content is protected !!