Applied linear operators and spectral methods/Lecture 1

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Linear operators can be thought of as infinite dimensional matrices. Hence we can use well known results from matrix theory when dealing with linear operators. However, we have to be careful. A finite dimensional matrix has an inverse if none of its eigenvalues are zero. For an infinite dimensional matrix, even though all the eigenvectors may be nonzero, we might have a sequence of eigenvalues that tend to zero. There are several other subtleties that we will discuss in the course of this series of lectures.

Let us start off with the basics, i.e., linear vector spaces.

Linear Vector Spaces ()[edit | edit source]

Let be a linear vector space.

Addition and scalar multiplication[edit | edit source]

Let us first define addition and scalar multiplication in this space. The addition operation acts completely in while the scalar multiplication operation may involved multiplication either by a real (in ) or by a complex number (in ). These operations must have the following closure properties:

  1. If then .
  2. If (or ) and then .

And the following laws must hold for addition

  1. = Commutative law.
  2. = Associative law.
  3. such that Additive identity.
  4. such that Additive inverse.

For scalar multiplication we have the properties

  1. .
  2. .
  3. .
  4. .
  5. .

Example 1: tuples[edit | edit source]

The tuples with

form a linear vector space.

Example 2: Matrices[edit | edit source]

Another example of a linear vector space is the set of matrices with addition as usual and scalar multiplication, or more generally matrices.

Example 3: Polynomials[edit | edit source]

The space of -th order polynomials forms a linear vector space.

Example 4: Continuous functions[edit | edit source]

The space of continuous functions, say in , also forms a linear vector space with addition and scalar multiplication defined as usual.

Linear Dependence[edit | edit source]

A set of vectors are said to be linearly dependent if not all zero such that

If such a set of constants do not exists then the vectors are said to be linearly independent.

Example[edit | edit source]

Consider the matrices

These are linearly dependent since .

Span[edit | edit source]

The span of a set of vectors is the set of all vectors that are linear combinations of the vectors . Thus

where

as vary.

Spanning set[edit | edit source]

If the span = then is said to be a spanning set.

Basis[edit | edit source]

If is a spanning set and its elements are linearly independent then we call it a basis for . A vector in has a unique representation as a linear combination of the basis elements. why is it unqiue?

Dimension[edit | edit source]

The dimension of a space is the number of elements in the basis. This is independent of actual elements that form the basis and is a property of .

Example 1: Vectors in [edit | edit source]

Any two non-collinear vectors is a basis for because any other vector in can be expressed as a linear combination of the two vectors.

Example 2: Matrices[edit | edit source]

A basis for the linear space of matrices is

Note that there is a lot of nonuniqueness in the choice of bases. One important skill that you should develop is to choose the right basis to solve a particular problem.

Example 3: Polynomials[edit | edit source]

The set is a basis for polynomials of degree .

Example 4: The natural basis[edit | edit source]

A natural basis is the set where the th entry of is

The quantity is also called the Kronecker delta.

Inner Product Spaces[edit | edit source]

To give more structure to the idea of a vector space we need concepts such as magnitude and angle. The inner product provides that structure.

The inner product generalizes the concept of an angle and is defined as a function

with the properties

  1. overbar indicates complex conjugation.
  2. Linear with respect to scalar multiplication.
  3. Linearity with respect to addition.
  4. if and if and only if .

A vector space with an inner product is called an inner product space.

Example 1:[edit | edit source]

Example 2: Discrete vectors[edit | edit source]

In with and the Eulidean norm is given by

With the standard norm is

Example 3: Continuous functions[edit | edit source]

For two complex valued continuous functions and in we could approximately represent them by their function values at equally spaced points.

Approximate and by

With that approximation, a natural norm is

Taking the limit as (show this)

If we took non-equally spaced yet smoothly distributed points we would get

where is a smooth weighting function (show this).

There are many other inner products possible. For functions that are not only continuous but also differentiable, a useful norm is

We will continue further explorations into linear vector spaces in the next lecture.

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