pre-hilbert spaces

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Wednesday September 01, 2021

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LOGISTICSLOGISTICSSelf assessment

Due today (firm deadline)Upload on Gradescope

Assignment 2 coming

Graduate teaching assistants + office hours

Monday 1:00pm - 2:00pm: TSRB 423 (Shi-Yuan)Tuesdays 12:00pm - 1:00pm: Tutorial Lab in Van Leer room C449 Cubicle D (Kayla)Wednedays 5:30 pm - 6:30pm: virtual (Kayla)Thursdays 10am - 11 am: Tutorial Lab in Van Leer room C449 Cubicle B (Kayla)Friday 1:00pm - 2:00pm: TSRB 423 (Meng-Che)

Course slides: https://bloch.ece.gatech.edu/teaching/ece7750fa21/

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WHAT’S ON THE AGENDA FOR TODAY?WHAT’S ON THE AGENDA FOR TODAY?Last time:

Formal introduction to vector spacesKey take-way: understand that vectors are not just arrows in (functions)Key concepts: subspaces, spans, linear combinationsQuestions?

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WHAT’S ON THE AGENDA FOR TODAY?WHAT’S ON THE AGENDA FOR TODAY?Last time:

Formal introduction to vector spacesKey take-way: understand that vectors are not just arrows in (functions)Key concepts: subspaces, spans, linear combinationsQuestions?

Today: pre-Hilbert spaces (inner product spaces)

More on vector spaces: linear independence, bases, dimensionNorm, inner product

Monday September 06, 2021: Hilbert spaces

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LINEAR INDEPENDENCELINEAR INDEPENDENCELet be a set of vectors in a vector space

Definition. (linear combination)

is linearly independent (or the vectors are linearly independent ) if (and only if)

Otherwise the set is (or the vectors are) linearly dependent.

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LINEAR INDEPENDENCELINEAR INDEPENDENCELet be a set of vectors in a vector space

Definition. (linear combination)

is linearly independent (or the vectors are linearly independent ) if (and only if)

Otherwise the set is (or the vectors are) linearly dependent.

Example.

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LINEAR INDEPENDENCELINEAR INDEPENDENCELet be a set of vectors in a vector space

Definition. (linear combination)

is linearly independent (or the vectors are linearly independent ) if (and only if)

Otherwise the set is (or the vectors are) linearly dependent.

Example.

Example.

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LINEAR INDEPENDENCELINEAR INDEPENDENCELet be a set of vectors in a vector space

Definition. (linear combination)

is linearly independent (or the vectors are linearly independent ) if (and only if)

Otherwise the set is (or the vectors are) linearly dependent.

Example.

Example.

Proposition.

Any set of linearly dependent vectors contains a subset of linearly independent vectors with the same span.

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BASESBASESDefinition. (basis)

A basis of vector subspace of a vector space is a countable set of vectors such that:

1. 2. is linearly independent

Definition. (dimension)If a non vector space as a finite basis with elements, is called the dimension of ,denoted . If the basis has an infinite number of elements, the dimension is infinite

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BASESBASESDefinition. (basis)

A basis of vector subspace of a vector space is a countable set of vectors such that:

1. 2. is linearly independent

Definition. (dimension)If a non vector space as a finite basis with elements, is called the dimension of ,denoted . If the basis has an infinite number of elements, the dimension is infinite

Proposition. Any two bases for the same finite dimensional vector space contain the same number ofelements.

You should be somewhat familiar with bases (at least in ):

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BASESBASESDefinition. (basis)

A basis of vector subspace of a vector space is a countable set of vectors such that:

1. 2. is linearly independent

Definition. (dimension)If a non vector space as a finite basis with elements, is called the dimension of ,denoted . If the basis has an infinite number of elements, the dimension is infinite

Proposition. Any two bases for the same finite dimensional vector space contain the same number ofelements.

You should be somewhat familiar with bases (at least in ):

the representation of a vector on a basis is uniqueevery subspace has a basishaving a basis reduces the operations on vectors to operations on their components

Things sort of work in infinite dimensions, but we have to be bit more careful

Example. ,

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NORMNORMThe properties of vector space seen thus far provide an algebraic structure

We are missing a topological structure to measure length and distance

Definition. (Norm)A norm on a vector space over is a function that satisfies:

Positive definiteness: with equality iff Homogeneity: Subadditivity:

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NORMNORMThe properties of vector space seen thus far provide an algebraic structure

We are missing a topological structure to measure length and distance

Definition. (Norm)A norm on a vector space over is a function that satisfies:

Positive definiteness: with equality iff Homogeneity: Subadditivity:

measures a length, measures a distance

Example (Norms (find the imposter))

?

Example (More examples)

See board

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INNER PRODUCTINNER PRODUCTIn addition to a topological and algebraic strucure, what if we want to do geometry?

Definition. (Inner product and inner product space)

An inner product space over is a vector space equipped with a positive definite symmetric bilinear form called an inner product

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INNER PRODUCTINNER PRODUCTIn addition to a topological and algebraic strucure, what if we want to do geometry?

Definition. (Inner product and inner product space)

An inner product space over is a vector space equipped with a positive definite symmetric bilinear form called an inner product

An inner product space is also called a pre-Hilbert space

Example (More examples)

See board

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INNER PRODUCTINNER PRODUCTIn addition to a topological and algebraic strucure, what if we want to do geometry?

Definition. (Inner product and inner product space)

An inner product space over is a vector space equipped with a positive definite symmetric bilinear form called an inner product

An inner product space is also called a pre-Hilbert space

Example (More examples)

See board

Unless stated otherwise, we will only deal with vector spaces on (things don’t change too much on butone should be a bit more careful)

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INDUCED NORM AND ORTHOGONALITYINDUCED NORM AND ORTHOGONALITYProposition (Induced norm)

In an inner product space, an inner product induces a norm

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INDUCED NORM AND ORTHOGONALITYINDUCED NORM AND ORTHOGONALITYProposition (Induced norm)

In an inner product space, an inner product induces a norm

Theorem (Parallelogram law)

A norm is induced by an inner product on iff

If this is the case, the inner product is given by the polarization identity

Proposition (Cauchy-Schwartz inequality)

An inner product satisfies

Definition. (Orthogonality)

Two vectors are orthogonal if . We write for simplicity.

A vector is orthogonal to a set if . We write for simplicity.

Theorem (Pythagorean theorem) If then

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