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MAT223H1

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MAT223H1b.doc Lecture #11 Tuesday, February 10, 2004 O RTHOGONALITY Definition Let X, Y be vectors in R . The dot product isY = X Y R . x1 y1 x2 y 2 n So, ifX = , Y = , X = x1 1+ x2y 2+ + xny n = xjy j. j=1 x y n n Definition x1 The lengthof a vectorX, denoted X = X X . So ifX = x 2, X = x 2 + x 2++ x 2 0 . 1 2 n x n Example 1 1 0 2 Let X = , Y = . Find X Y and X . 2 3 4 2 X Y = 1)(1 + 0 2) + (2 3( +(4) 2 = 15. X = (1) + 0 ( ) 2 ( + 4)2 = (2) . Theorem n Let X, Y, Z be vectors Rn. Then: n n 1) X Y = Y X . (Proof xjy j= y j j) =1 j=1 2) X (Y +Z )= X Y + X Z . 3) For a R , (aX Y = X aY) = a X Y . 4) X 0 , X = 0 x = 0 . Example Let X be a vector iRn, X 0. Find all R nsuch that Y is collinear to X and is unitary. 1 1 Let Y = aX,aR . Y = aX = a X . We want Y =1 , so a X =1 a = X a = X . 1 So, Y = X . X Page 1 of 51 www.notesolution.com MAT223H1b.doc Definition n Let X, Y be two vectors in R. X and Y are orthogonal iff X Y = 0 . { } n { } Let X 1 X 2 , X k be a set of vectors in . X 1 X 2 , Xk is orthogonaliff xi 0,i =1,2, ,k and X X = 0,i j. i j Moreover, if X j =1, j =1,2,, k , then X 1 X2,, X k is orthonormal. Example 1 0 0 The standard basis ofR n is a orthonormal set.E = , E = , E = . 1 2 3 Definition If {X1, X2,, X k} is an orthogonal s, then {a1X 1a 2 ,2,a X k k } is also an orthogonal seif a 0, j =1,2,,k . j 1 For a j , the set{a1X 1a 2 ,2,a X k k } will be orthonormal. X j Example 2 0 1 Let X 1 = , X 2 = 3 , X3 = 1 . Construct an orthonormal set. 1 3 1 {X1, X2, X 3} is an orthogonal set because: X 0 . i X1X 2 = 0 , X 1 =30, X X =20 3 Find a j,j =1,2,3 such that a 1 1a X2,a2X 3 3} is orthonormal: 1 1 1 1 1 a1= = = , a 2 , a 3 . X1 2 ) + 1 ( ) 1 2 ( ) 6 3 2 3 X X X So, 1 , 2 , 3 is orthonormal. 6 3 2 3 Theorem (Pythagorean) If X and Y are orthogonal, then X +Y 2= X 2 + Y 2 . Proof X and Y are orthogonal; it means X Y = 0 . So, 2 2 2 X +Y = X +Y X +Y = X X + X Y +Y X +Y Y = X X +0+0+Y Y = X + Y . Page 2 of 51 www.notesolution.com MAT223H1b.doc Theorem Every orthogonal set of vectors ofR nis independent. Proof Let {X , X , , X } be an orthogonal set. Consider Y = t X + t X + t X = 0 . We want to prove 1 2 k 1 1 2 2 k k that t j 0 . X jY = 0 X jt X1+t1X +2+t2X k k)= t1X 1 X jt X2 2 ++j X X +j+tjX j k k= 0 . t X 2 = 0 t = 0 j j j Theorem If {E , E , , E } is an orthogonal basis oRn, then X R , n 1 2 n X 1 X E 2 X n X = E 1+ E 2++ En . E 2 E 2 E 2 1 2 n Proof n n Let X R . Then X = t1 1+t E2+2+t E n n because is a {E 1,E2 ,E n} basis ofR . 2 X E j X E j t E1 1 E +2t2E E =nt n E j+t1 1E +j+t E Ej= j E j n n j j j tj= 2 . E j Example 0 1 3 Let X1 = , X 2 = 3, X 3= 1. Show that {X 1,X 2,X 3} is an orthogonal basis of . 0 3 1 It is enough to prove X 1 X 2 0, X 2 X 3 0 , X 1 X 3 0 because: {X ,X , X } is an orthogonal set, sX ,X ,X are independent. 1 2 3 1 2 3 dim R = 3. So a set of 3 linearly independent vectors is a basis. Example 0 1 Write X = b as a linear combination of X = 1 , X = 3 , X = 1 . 1 2 3 3 1 Since {X ,X ,X } is orthogonal, 1 2 3 X X1 X X2 X X3 2a + 3b + a + X = 2 X1+ 2 X 2 2 X 3 = X1 + X 2+ X 3. X1 X 2 X 3 5 6 3 Page 3 of 51 www.notesolution.com

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