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MATH 136 (145)
Lecture

# lect136_4_w14.pdf

5 Pages
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School
University of Waterloo
Department
Mathematics
Course
MATH 136
Professor
Robert Andre
Semester
Spring

Description
Monday, January 13 − Lecture 4 : Dot products and norms Concepts: n 1. dot product of two vectors in ℝ . 2. algebraic properties of dot product of two vectors. 3. length or norm of a vector in ℝ . n 4. unit vector in the direction of a n 5. angle between two vectors in ℝ . 6. distance between two points in ℝ . n 7. Cauchy-Schwartz inequality n 8. orthogonal vectors or ortnogonal sets in ℝ . 9. orthonormal sets in ℝ . 4.1 Dot product operation on ℝ definition. Operation What we start with : What we get : n 1 number x⋅y in ℝ : 2 vectors x and y in ℝ : x⋅y = x y + x y + ..., x y . x = (x 1 x2, ...,nx ) 1 1 2 2 n n Dot product and Other notation for dot-product: y = ( y , y , ...,y ). x⋅y = < x, y > 1 2 n 4.1.1 Examples − If x = (2, 1, 7) and y = (3, − 9, 0) verify that x ⋅ y = −3. n 4.1.2 Theorem − Algebraic properties of dot product. Suppose x, y and z are in ℝ and α in ℝ. Then DP1 x ⋅x ≥ 0 DP2 x ⋅y = y ⋅x DP3 (αx) ⋅y = α(x ⋅y) DP4 x ⋅y + z) = x ⋅y + x ⋅z Proof is straight forward and is left as an exercise. Norm or length of a vector 4.2 Definition − For a vector x in ℝ , the length or norm of x, denoted by || x || , is the number n 4.2.1 Properties of the norm in ℝ . 3 It can be shown that the norm on ℝ satisfies 3 properties: N1 For every vector x, || x || ≥ 0. Furthermore, ||x|| = 0 ⇔ x = 0. (Positive definite property) N2 || αx || = |α| || x || for all scalars α, and x in V. (Homogeneous property) N3 || x + y || ≤ || x || + || y || for all x, y in V(Triangle inequality property) n 4.2.2 Observation − If a is any non-zero vector in ℝ , then the vector defined as b = (1/|| a ||)a has norm equal to 1 since The vector a/|| a || is called the unit vector in the direction of a. Note that if b is a vector which points in the same direction as a then This is true since if b and a point in the same direction then they both have the same unit vector, i.e., a / || a || = b / || b ||. 4.2.3 Remark – In ℝ the norm, thus defined, leads to the familiar distance formula between two points x and y. The distance betweenx = (x , x ) and y = (y , y ) is 1 2 1 2 n Definition − Let a and b be two vectors in ℝ . The distance between a and b is defined as || b − a || n 4.3 Theorem − Cauchy-Schwartz inequality. For any vectors x and y in ℝ | x⋅y | ≤ || x || || y || Proof : n Let x, y be two vectors in ℝ and t be a variable scalar. For any real number t: (Be sure to verify the equality between the expression in the f
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The text passage is a set of class notes from a Mathematics course, specifically MATH 136, and it covers various concepts related to dot products and norms in vector spaces. It includes definitions, properties, and theorems, along with some examples and proofs.
no_question

the passage does not contain any explicit mathematical questions or problems that require solving. it is primarily expository, providing definitions, properties, and theorems related to the dot product and norms in vector spaces.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Class Notes (809,510)
Mathematics (1,874)
MATH 136 (145)
Lecture

# lect136_4_w14.pdf

5 Pages
123 Views

School
University of Waterloo
Department
Mathematics
Course
MATH 136
Professor
Robert Andre
Semester
Spring

Description
Monday, January 13 − Lecture 4 : Dot products and norms Concepts: n 1. dot product of two vectors in ℝ . 2. algebraic properties of dot product of two vectors. 3. length or norm of a vector in ℝ . n 4. unit vector in the direction of a n 5. angle between two vectors in ℝ . 6. distance between two points in ℝ . n 7. Cauchy-Schwartz inequality n 8. orthogonal vectors or ortnogonal sets in ℝ . 9. orthonormal sets in ℝ . 4.1 Dot product operation on ℝ definition. Operation What we start with : What we get : n 1 number x⋅y in ℝ : 2 vectors x and y in ℝ : x⋅y = x y + x y + ..., x y . x = (x 1 x2, ...,nx ) 1 1 2 2 n n Dot product and Other notation for dot-product: y = ( y , y , ...,y ). x⋅y = < x, y > 1 2 n 4.1.1 Examples − If x = (2, 1, 7) and y = (3, − 9, 0) verify that x ⋅ y = −3. n 4.1.2 Theorem − Algebraic properties of dot product. Suppose x, y and z are in ℝ and α in ℝ. Then DP1 x ⋅x ≥ 0 DP2 x ⋅y = y ⋅x DP3 (αx) ⋅y = α(x ⋅y) DP4 x ⋅y + z) = x ⋅y + x ⋅z Proof is straight forward and is left as an exercise. Norm or length of a vector 4.2 Definition − For a vector x in ℝ , the length or norm of x, denoted by || x || , is the number n 4.2.1 Properties of the norm in ℝ . 3 It can be shown that the norm on ℝ satisfies 3 properties: N1 For every vector x, || x || ≥ 0. Furthermore, ||x|| = 0 ⇔ x = 0. (Positive definite property) N2 || αx || = |α| || x || for all scalars α, and x in V. (Homogeneous property) N3 || x + y || ≤ || x || + || y || for all x, y in V(Triangle inequality property) n 4.2.2 Observation − If a is any non-zero vector in ℝ , then the vector defined as b = (1/|| a ||)a has norm equal to 1 since The vector a/|| a || is called the unit vector in the direction of a. Note that if b is a vector which points in the same direction as a then This is true since if b and a point in the same direction then they both have the same unit vector, i.e., a / || a || = b / || b ||. 4.2.3 Remark – In ℝ the norm, thus defined, leads to the familiar distance formula between two points x and y. The distance betweenx = (x , x ) and y = (y , y ) is 1 2 1 2 n Definition − Let a and b be two vectors in ℝ . The distance between a and b is defined as || b − a || n 4.3 Theorem − Cauchy-Schwartz inequality. For any vectors x and y in ℝ | x⋅y | ≤ || x || || y || Proof : n Let x, y be two vectors in ℝ and t be a variable scalar. For any real number t: (Be sure to verify the equality between the expression in the f
More Less

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user
The text passage is a set of class notes from a Mathematics course, specifically MATH 136, and it covers various concepts related to dot products and norms in vector spaces. It includes definitions, properties, and theorems, along with some examples and proofs.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems that require solving. It is primarily expository, providing definitions, properties, and theorems related to the dot product and norms in vector spaces.