Section 6.3 d i E Orthogonal and orthonormal basis a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 211: Linear Algebra Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 1 / 13
Goal: d i E 1 Orthogonal and orthonormal basis. a 2 Coordinates relatives to orthonormal basis. l l u 3 Orthogonal Projection. d b 4 The Gram–Schmidt Process. A . r D Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 2 / 13
Normalizing Procedure d i E Goal: To create a unit vector v ′ from a given vector v . a l l 1 u v ′ = || v || v d b A Check that v ′ indeed is a unit vector! . r D Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 3 / 13
Orthogonal Set Definition 1 d A set of vectors { v 1 v 2 , . . . , v n } is called orthogonal set if � v i , v j � = 0 , for i E all i � = j . a A set if called orthonormal basis if it is orthogonal and each vector is a l l u unit vector. d b A Example 2 . 1 3 0 r D , v 2 = , v 3 = is an orthogonal set. Verify that v 1 = − 1 3 0 0 0 2 Find an orthonormal set. Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 4 / 13
Orthogonality implies linearly independent Theorem 3 If S = { v 1 , v 2 , . . . , v n } is an orthogonal set, then S is linearly independent. d i E a l l u d b A . r D Definition 4 A basis consisting of orthogonal vectors is called orthogonal basis . Similarly, a basis consisting of orthonormal vectors is called orthonormal basis . Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 5 / 13
Relative coordinates to orthonormal basis Theorem 5 (a) If S = { v 1 , v 2 , . . . , v n } is an orthogonal basis then u = c 1 v 1 + · · · + c n v n d i E with c i = � u , v i � || v i || 2 . a l (b) In case S is orthonormal basis, then c i = � u , v i � . l u d Proof: Consider || u + v || 2 . b A . r D Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 6 / 13
Example 6 − 3 4 0 5 5 4 , v 2 = 3 , v 3 = is an Verify that the vectors v 1 = 0 5 5 0 0 1 d orthogonal basis. Then write each of the following vectors as linear i E combination of v 1 , v 2 , v 3 . a 2 l l u (a) u = 1 d b − 2 A 1 . (b) u = 3 r D 4 1 7 − 3 (c) u = 7 5 7 Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 7 / 13
Orthogonal Projection Theorem 7 d Let W be a subspace of V , then each vector v ∈ V can be written in i E exactly one way as a v = w + ˆ w l l u w ∈ W ⊥ . d where w ∈ W and ˆ b A . r D The vector w above is called orthogonal projection of u on W and denoted by w = proj W u . The vector ˆ w above is called orthogonal projection of u on W ⊥ and denoted by w = proj ⊥ W u . Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 8 / 13
How to calculate proj W u First we find an orthogonal basis for W , say { v 1 , v 2 , . . . , v n } . Then, proj W u = c 1 v 1 + c 2 v 2 + · · · + c n v n d i with c i = � u , v i � E || v i || 2 a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 9 / 13
Calculate the projections Example 8 − 1 1 1 − 1 Let v 1 = , v 2 = be a basis for a subspace W . Find proj W u , d − 1 1 i E 1 1 a 1 l l u 2 where u = . d 0 b A − 2 . r D Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 10 / 13
Calculate the projections Example 9 1 4 0 2 Let v 1 = , v 2 = be a basis for a subspace W . Find proj W u , d − 3 1 i E − 1 1 a 1 l l u 2 where u = . d 0 b A − 2 . r D Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 11 / 13
Creating orthogonal basis from any basis Definition 10 If W is a subspace of an inner vector space V , then the set of all vectors in V that are orthogonal to every vector in W is called the orthogonal d complement of W and is denoted by W ⊥ . i E a l W ⊥ : = { ˆ l w ∈ V | � ˆ w , w � = 0, for all w ∈ W } u d b A . r D Theorem 11 1 W ⊥ is a subspace of W . 2 W ∩ W ⊥ = { 0 } W ⊥ � ⊥ � 3 Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 12 / 13
Row space and null space are orthogonal Example 12 Let W = span { w 1 , w 2 , w 3 } where, 2 − 1 d 4 i , w 2 = , w 3 = 1 − 4 E − 5 w 1 = 3 2 13 a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 13 / 13
Row space and null space are orthogonal Example 13 Let W = span { w 1 , w 2 , w 3 } where, 3 − 1 d 4 i 0 − 2 E 2 , w 2 = , w 3 = w 1 = 1 − 2 3 a l − 2 l − 3 1 u d b A . r D Dr. Abdulla Eid (University of Bahrain) Orthogonal and orthonormal 14 / 13
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