Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by a linear model.
We will begin our studies by studying systems of linear equations. Without becoming to formal in the notation and language we will study the basic properties of the solutions for homogeneous and non homogeneous systems of linear equations. We will get known to the Gaussian algorithm for solving systems of linear equations. This algorithm will re-occur repeatedly in this lecture note. Towards the end of this first chapter two problems will in a natural way catch our attention. In order to solve them we need to begin to formalize the observations we made so far.
The formalization will begin by extracting the essential properties of the numbers we have used in the first chapter. This will lead to the concept of a field (Definition 2.1). Next we will formalize the properties which the solutions of a homogeneous system of equations possesses: the sum of two solutions of a homogeneous system of equations is again a solution this system of equations, and the same is true if we multiply a solution of such a system of equations by a number (that is an element of a field). This will lead to the concept of a vector space (Definition 2.4). Roughly spoken a vector space over a field is a set V where we can form sums of arbitrary elements and where we can multiply any element by scalars of the field and such that this addition and scalar multiplication satisfies certain rules which seem natural to us.
In the third chapter we will then start to study the relation ship between. We will introduce the concept of a linear map between vector spaces (Definition 3.1). The whole chapter is devoted to the study of these kind of maps. One of the main theme of this chapter will be the matrix description of linear maps between finite dimensional vector spaces. We will explore the relation ship between matrices and linear maps and what we can all conclude from that.
The fourth chapter will then be devoted to the first studies of determinant functions. First we introduce the new concept and show what hypothetical properties a determinant function would have. It will turn out that the determinant function -- in case it exists -- must be unique. This will be the reason why we will later be able to give an affirmative answer about the uniqueness of the above decomposition. But we will first have to show that a determinant function exists and this is done in the second section of the chapter about determinants. When we are finally convinced about the existence of determinants we will study in the remaining part of the chapter some basic properties of the determinant function. The chapter will finish with the presentation of the Leibniz formula which shows the beauty and symmetries of the determinant function (Theorem 4.28 ).
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