Math 127C Homework 1
(1) (Triangle Inequality) [Exercise 1.1.(b)] Prove that
||x + y|| ≤||x||+||y||.
[Hint: Compute hx + y, x + yi and apply the Cauchy-Schwarz inequality which says that hx, yi ≤||x|| ||y||.]
(2) (Matrix supremum norm)[Exercise 1.2] If A is an r by m matrix and B is an m by c matrix show that
|AB| ≤ m|A||B|.
(3) (Theorem 18.3) Find a shortest sequence of type (2) and type (3) elementary row operations which have the effect of switching the first two rows of a matrix. Show that there is no such sequence using only type (2) operations.
(4) (Theorem 1.6) Prove that if B is the matrix obtained by applying an elementary row operation to A, then
rank B = rank A.