Description
Instructions: Read textbook pages 59 to 62 before working on the homework problems. Show all steps to get full credits.
1. Find the matrix for the linear transformation which reflects every 2-dimensional
vector across the y axis and hen rotate by an angle of π/4.
2. Is (A + B)
2 = A2 + 2AB + B2
true for two square matrices A, B of the same
sizes? Justify your answer.
3. Let A, B be matrices of appropriate sizes, prove that (AB)
∗ = B∗A∗
.
4. Let A, B be two square upper-triangular matrices with the same size, prove
that AB is also upper-triangular. The same conclusion applies for lowertriangular matrices.
5. Let F = R or C, prove that A ∈ F
n×n
is invertible if and only if all the
columns of A are linearly independent. (Hint: to show A is invertible, it is
enough to show that there exists a matrix B such that AB = I as BA = I
will follow from AB = I.