Exercises on orthogonal matrices and Gram-Schmidt
Problem 17.1: (4.4 #10.b Introduction to Linear Algebra: Strang) Orthonormal vectors are automatically linearly independent.
Matrix Proof: Show that Qx = 0 implies x = 0. Since Q may be rectangular, you can use QT but not Q−1.
Problem 17.2: (4.4 #18) Given the vectors a, b and c listed below, use the Gram-Schmidt process to find orthogonal vectors A, B, and C that span the same space.
a = (1, −1, 0, 0), b = (0, 1, −1, 0), c = (0, 0, 1, −1).
Show that {A, B, C} and {a, b, c} are bases for the space of vectors perpendicular to d = (1, 1, 1, 1).
