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 ∈ Fn×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.
