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Inner Product Spaces, Thesis of Linear Algebra

Ho Chi Minh City University of Technology (HCMCUT)Linear Algebra

An in-depth exploration of inner product spaces, a fundamental concept in linear algebra and functional analysis. It covers the definition of real inner product spaces, including the axioms that must be satisfied, and introduces key related concepts such as the length (or norm) of a vector, the distance between two vectors, and the angle between two vectors. The document also delves into the topic of orthogonality, discussing orthogonal and orthonormal sets, the gram-schmidt process for constructing orthogonal bases, and orthogonal complements. Additionally, it explores the concept of orthogonal projections, explaining how to find the orthogonal projection of a vector onto a subspace and the distance between a vector and a subspace. Several illustrative examples and mathematical derivations to solidify the understanding of these fundamental ideas in linear algebra.

Typology: Thesis

2022/2023

Uploaded on 05/22/2023

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INNER PRODUCT SPACES

ELE CT RON IC V ER SI ON O F LE CT UR E

Dr. Lê Xuân Đại

HoChiMinh City Universityof Technology

Faculty of Applied Science, Department of Applied Mathematics

Email: [email protected]

HCMC — 2018.

Dr. Lê XuânĐại (HCMUT-OISP) INNER PRODUCT SPACES HCMC — 2018. 1 / 41

Discover Thesis of Linear Algebra Ho Chi Minh City University of Technology (HCMCUT)

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Download Inner Product Spaces and more Thesis Linear Algebra in PDF only on Docsity!

INNER PRODUCT SPACES ELECTRONIC VERSION OF LECTURE

Dr. Lê Xuân Đại

Faculty of Applied Science, Department of Applied Mathematics^ HoChiMinh City University of Technology Email: [email protected]

HCMC — 2018.

OUTLINE

(^1) REAL WORLD PROBLEMS

OUTLINE

(^1) REAL WORLD PROBLEMS (^2) REAL INNER PRODUCT SPACE (^3) ORTHOGONALITY

OUTLINE

(^1) REAL WORLD PROBLEMS (^2) REAL INNER PRODUCT SPACE (^3) ORTHOGONALITY (^4) MATLAB

Real world problems WORK DONE BY A FORCE −→ F

W = →− F .−→ s = F. s. cos α

Real world problems

−→ a = ( a 1 , a 2 ), −→ b = ( b 1 , b 2 ).

< −→ a , →− b >= a 1. b 1 + a 2. b 2 ; ||−→ a || =

a^21 + a 22

cos α = <

→− a , −→ b > ||→− a ||.||→− b ||

; d (−→ a , →− b ) = ||−→ a − −→ b ||

Real inner product space Definition

A real vector space V is called a real

Euclidean inner product space if

< ·, · >: V × V → R

( x , y ) 7 −→< x , y > − which is called

inner product of 2 vectors.

The following axioms are satisfied

Real inner product space Definition

A real vector space V is called a real

Euclidean inner product space if

< ·, · >: V × V → R

( x , y ) 7 −→< x , y > − which is called

inner product of 2 vectors.

The following axioms are satisfied

(^1) < x , y >=< y , x >, ∀ x , y ∈ V

Real inner product space Definition

A real vector space V is called a real

Euclidean inner product space if

< ·, · >: V × V → R

( x , y ) 7 −→< x , y > − which is called

inner product of 2 vectors.

The following axioms are satisfied

(^1) < x , y >=< y , x >, ∀ x , y ∈ V (^2) < x + y , z >=< x , z > + < y , z >, ∀ x , y , z ∈ V (^3) < αx , y >= α < x , y >, ∀ x , y ∈ V , ∀ α ∈ R.

Real inner product space Definition

A real vector space V is called a real

Euclidean inner product space if

< ·, · >: V × V → R

( x , y ) 7 −→< x , y > − which is called

inner product of 2 vectors.

The following axioms are satisfied

(^1) < x , y >=< y , x >, ∀ x , y ∈ V (^2) < x + y , z >=< x , z > + < y , z >, ∀ x , y , z ∈ V (^3) < αx , y >= α < x , y >, ∀ x , y ∈ V , ∀ α ∈ R.

4 < x , x >> 0, x 6 = 0 and < x , x >= 0 ⇔ x = 0

Real inner product space Definition

EXAMPLE 2.

On R 3 we define the standard inner product

( x , y ) 7 −→< x , y >= x 1. y 1 + x 2. y 2 + x 3. y 3 = x. yT

where x = ( x 1 , x 2 , x 3 ), y = ( y 1 , y 2 , y 3 ).

EXAMPLE 2.

On R n we define the standard inner product

( x , y ) 7 −→< x , y >= ∑ n i = 1^ xiyi^ =^ x. y

where x = ( x 1 , x 2 ,... , xn ), y = ( y 1 , y 2 ,... , yn ).

Real inner product space Definition

EXAMPLE 2.

On R 2 we define the weighted Euclidean

inner product of 2 vectors

( x , y ) 7 −→< x , y >= x 1. y 1 + 2 x 2. y 2

where x = ( x 1 , x 2 ), y = ( y 1 , y 2 ).

Real inner product space Definition

EXAMPLE 2.

On R 2 we define the weighted Euclidean

inner product of 2 vectors

( x , y ) 7 −→< x , y >= x 1. y 1 + 2 x 2. y 2

where x = ( x 1 , x 2 ), y = ( y 1 , y 2 ).

< x , y >= x 1. y 1 + 2 x 2. y 2 = y 1. x 1 + 2 y 2. x 2 =< y , x > < x + y , z >= ( x 1 + y 1 ) z 1 + 2( x 2 + y 2 ) z 2 = ( x 1 z 1 + 2 x 2 z 2 ) + ( y 1 z 1 + 2 y 2 z 2 ) =< x , z > + < y , z >

Real inner product space Definition

EXAMPLE 2.

On R 2 we define the weighted Euclidean

inner product of 2 vectors

( x , y ) 7 −→< x , y >= x 1. y 1 + 2 x 2. y 2

where x = ( x 1 , x 2 ), y = ( y 1 , y 2 ).

< x , y >= x 1. y 1 + 2 x 2. y 2 = y 1. x 1 + 2 y 2. x 2 =< y , x > < x + y , z >= ( x 1 + y 1 ) z 1 + 2( x 2 + y 2 ) z 2 = ( x 1 z 1 + 2 x 2 z 2 ) + ( y 1 z 1 + 2 y 2 z 2 ) =< x , z > + < y , z > < αx , y >= α. x 1. y 1 + 2 α. x 2. y 2 = α ( x 1 y 1 + 2 x 2 y 2 ) = α. < x , y >

Inner Product Spaces, Thesis of Linear Algebra

Related documents

Partial preview of the text

Download Inner Product Spaces and more Thesis Linear Algebra in PDF only on Docsity!

Dr. Lê Xuân Đại

HCMC — 2018.

A real vector space V is called a real

Euclidean inner product space if

< ·, · >: V × V → R

( x , y ) 7 −→< x , y > − which is called

inner product of 2 vectors.

The following axioms are satisfied

A real vector space V is called a real

Euclidean inner product space if

< ·, · >: V × V → R

( x , y ) 7 −→< x , y > − which is called

inner product of 2 vectors.

The following axioms are satisfied

A real vector space V is called a real

Euclidean inner product space if

< ·, · >: V × V → R

( x , y ) 7 −→< x , y > − which is called

inner product of 2 vectors.

The following axioms are satisfied

A real vector space V is called a real

Euclidean inner product space if

< ·, · >: V × V → R

( x , y ) 7 −→< x , y > − which is called

inner product of 2 vectors.

The following axioms are satisfied

4 < x , x >> 0, x 6 = 0 and < x , x >= 0 ⇔ x = 0

EXAMPLE 2.

On R 3 we define the standard inner product

where x = ( x 1 , x 2 , x 3 ), y = ( y 1 , y 2 , y 3 ).

EXAMPLE 2.

On R n we define the standard inner product

where x = ( x 1 , x 2 ,... , xn ), y = ( y 1 , y 2 ,... , yn ).

EXAMPLE 2.

On R 2 we define the weighted Euclidean

inner product of 2 vectors

where x = ( x 1 , x 2 ), y = ( y 1 , y 2 ).

EXAMPLE 2.

On R 2 we define the weighted Euclidean

inner product of 2 vectors

where x = ( x 1 , x 2 ), y = ( y 1 , y 2 ).

EXAMPLE 2.

On R 2 we define the weighted Euclidean

inner product of 2 vectors

where x = ( x 1 , x 2 ), y = ( y 1 , y 2 ).