Applied linear operators and spectral methods/Lecture 1

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 (S) edit

Let   be a linear vector space.

Addition and scalar multiplication edit

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: n tuples edit

The   tuples   with


form a linear vector space.

Example 2: Matrices edit

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

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


Example 4: Continuous functions edit

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

Linear Dependence edit

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

Consider the matrices


These are linearly dependent since  .

Span edit

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




as   vary.

Spanning set edit

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

Basis edit

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

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 R2 edit

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

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

The set   is a basis for polynomials of degree  .

Example 4: The natural basis edit

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


The quantity   is also called the Kronecker delta.

Inner Product Spaces edit

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


Example 2: Discrete vectors edit

In   with   and   the Eulidean norm is given by


With   the standard norm is


Example 3: Continuous functions edit

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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