Linear problems occur throughout mathematics and its applications. This course will examine the mathematical tools needed to study such problems. The simplicity and elegance of these tools is part of what makes linearity such a desirable property. Indeed, many problems that are not inherently linear are often made linear as a first step in their analysis.

Problems that require only a finite number of pieces of information to specify them can be reduced to the study of matrices and coordinate vectors. However other problems, such as the evolution of a sound wave, require so-called infinite dimensional analysis. In this course we shall study both.

The topics that we shall cover include:

• Approximation and metric spaces
• Linear transformations and their characteristics
• Infinite dimensional linear spaces

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