Solutions to ExercisesCollege of the RedwoodsMathematics DepartmentMath Math 45 — Linear AlgebraQuiz #7—Linear AlgebraDavid ArnoldCopyrightc 2000 [email protected] Revision Date: October 29, 2001 Version 1.00Directions: Place your solution in the space provided. No calculators al-lowed!Exercise 1. Consider the independent vectorsa1=1000, a2=1100, and a3=1101.Use the Gram-Schmidt algorithm to find an orthonormal set of vectors {q1, q2, q3}that span the same space as the vectors {a1, a2, a3}.Solutions to ExercisesExercise 1.• Step 1.LetA1= a1.A1=1000.• Step 2.A2= a2−a2· A1A1· A1A1=1100−1100·10001000·10001000=1100−111000=0100• Step 3.A3= a3−a3· A1A1· A1A1−a3· A2A2· A2A2=1101−1101·10001000·10001000−1101·01000100·01000100=1101−111000−110100=0001.Note that A1, A2,andA3already have length one (what a nice teacher),so our orthonormal set is{q1,q2,q3} =100001000001.Exercise