Homework #1 Solution

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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 ‖ ‖ ≤ ‖ ‖ ‖ ‖ .

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