Consider the following matrices A. The columns of the Q are obtained from A by applying the Gram-Schmidt Orthogonalization Process. Find a QR factorization of A. Check your answers. (a) A [ l l Q = [1''{5 2/Vs -2Vsl 1/Vs (b) A Q [ ; 3 5 J [ 5/6 -1/6 ] 1/6 5/6 -3/6 1/6 1/6 3/6 c) A 0 1 [ t/v'3 1/Vl -t/v'6 ] Q = 11V3 -1/Vl -11v'6 11V3 0 21v'6 (d) A [I ,J [ t/v'6 1/] 21v'6 Q -1v'6 11V3 11V3 [ 1 ll (e) A 5 -1 [3/v'i3 -21v'i4 -21v'i82 l Q = 2/ 3/v'i4 3/\/'i82 -11v'i4 13 Jv'i82

Lecture 1: Introduction Simplify the Expressions: 1. 5(3x+4)-4 = 15x+20-4 = 15x+16 2. 4(5y-3)-(6y+3) = 20y-12-6y-3 = 24y-15 3. l -5-3 l = l-8 l = 8 Order of Operations Please Excuse My Dear Aunt Sally = Parenthesis Exponents Multiplication Division Addition Subtraction Exponents: 3 2 5 1. (x 1/3x 2/3 x 1 2. (x ) (x ) = x =x 3. (x ) = x 12 4. y / y = y 3 0 5. z =-2 2 6. ¼ = 4 = 16 7. 1/(-x) = (-x)2 8. 7x y = 7y /x 5 9. (-3x y ) = 81x y 12 Square Roots: 1. √81 = 9 2 2. √81x = 9x 3. √100/9 = 10/3 4. √500 = √5(100) = 10 √5 5. √ 48 x /√6 x = √8x 2 = 2x √2 6. 5 √x+7 x√12√ x 7√ √ 12=7 3+√ 3=9√√ 7. 8. 1/ √ 7= √/7