d i E Inner Product a l l u d Dr. Abdulla Eid b A College - PowerPoint PPT Presentation

Section 6.1 d i E Inner Product 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) Inner Product 1 / 13

Goal: d i 1 Cauchy Schwarz Inequality. E 2 Angle between vectors. a l l 3 Properties of length and distance. u d 4 Orthogonality. b A 5 Orthogonal Complement. . r D Dr. Abdulla Eid (University of Bahrain) Inner Product 2 / 13

Cauchy Schwarz Ineqality Theorem 1 If u , v are two vectors, then d | � u , v � | ≤ || u |||| v || i E Proof: Let a = || u || 2 , b = 2 � u , v � , and c = � v , v � . Consider a l � t u + v , t u + v � ≥ 0 so the discriminant is less than or equal to zero. l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inner Product 3 / 13

Consequences of Cauchy Schwarz Inequality | � u , v � | ≤ || u |||| v || | � u , v � | || u |||| v || ≤ 1 d i E − 1 ≤ � u , v � a || u |||| v || ≤ 1 l l u d � u , v � cos θ = || u |||| v || and 0 ≤ θ ≤ π b A . r D Definition 2 The angle θ between u and v is defined by � � u , v � � θ = cos − 1 || u |||| v || Dr. Abdulla Eid (University of Bahrain) Inner Product 4 / 13

Find the angle Example 3 Find the cosine of the angle between u and v , with the standard inner d product. i E     1 4 a 1 u =  and v = 2 4  . l   l u − 4 3 d b     0 5 A 1 2 2 u =      and v =  . .     2 − 3 r   D − 4 3 3 p = 1 + 3 X − 5 X 2 , q = 2 − 3 X 2 . � 2 � � 0 � 4 1 4 U = , V = − 2 3 2 0 Dr. Abdulla Eid (University of Bahrain) Inner Product 5 / 13

Properties of Length and Distance Theorem 4 If u , v , w are three vectors, then d || u + v || ≤ || u || + || v || ( Triangle inequality for vectors ) i E a l l u d ( u , v ) ≤ d ( u , w ) + d ( v , w ) ( Triangle inequality for distances ) d b A Proof: Consider || u + v || 2 . . r D Dr. Abdulla Eid (University of Bahrain) Inner Product 6 / 13

Orthogonality d Definition 5 i E Two vectors u , v are called orthogonal if � u , v � = 0. a l l u d b A . r D Question: What is the angle between any two orthogonal vectors? Dr. Abdulla Eid (University of Bahrain) Inner Product 7 / 13

Check for orthongonality Example 6 Determine whether u and v are orthogonal or not, with the standard inner d product. i E     1 4 a 1 u =  and v = 2 4  . l   l u 3 − 4 d b     0 5 A 1 2     2 u =  and v =  . .     − 3 2 r   D − 4 3 3 f = x , q = X 2 on [ − 1, 1 ] . � 1 � � 0 � 0 1 4 U = , V = 1 1 0 0 Dr. Abdulla Eid (University of Bahrain) Inner Product 8 / 13

Pythagorean Theorem Theorem 7 If u , v are orthogonal vectors, then d || u + v || 2 = || u || + || v || i E Proof: Consider || u + v || 2 = � u + v , u + v � . a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inner Product 9 / 13

Example 8 √ Show that if u , v are orthogonal unit vectors in V , then || u − v || = 2. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inner Product 10 / 13

Orthogonal Complement Definition 9 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 10 1 W ⊥ is a subspace of W . 2 W ∩ W ⊥ = { 0 } W ⊥ � ⊥ � 3 Dr. Abdulla Eid (University of Bahrain) Inner Product 11 / 13

Row space and null space are orthogonal Example 11 Let W = span { w 1 , w 2 , w 3 } where,       2 − 1 d 4 i  , w 2 =  , w 3 = w 1 = − 4 E − 5 1     3 2 13 a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inner Product 12 / 13

Row space and null space are orthogonal Example 12 Let W = span { w 1 , w 2 , w 3 } where,       3 − 1 d 4 i − 2 E 0 2       w 1 =  , w 2 =  , w 3 =       1 − 2 3 a     l − 2 l − 3 1 u d b A . r D Dr. Abdulla Eid (University of Bahrain) Inner Product 13 / 13