Vector spaces

Linear algebra is the study of linear structures. Historically, the motivation for the subject was solving systems of linear equations. Given an equation like

$$ \begin{cases} 3x + y = 5, \\ x + 3y = 7, \end{cases} $$

how do we solve it? This question led mathematicians such as Leibniz, Cramer, and Gauss to study properties of systems of linear equations. Later, the notion of matrices was introduced to record and manipulate equations more easily. For example, the above equation could be written as

$$ \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 7 \end{bmatrix} $$

using matrix notation. Then multiply the inverse matrix from the left on both sides to solve the equation.

But such a concrete point of view is not always helpful in developing a theory. When you know too much about an object, it is easy to get lost in the pile of information. Abstraction is supposed to deal with such problems by taking away some, but not too much, information so that only the essential ones stand out.

In the abstract approach to linear algebra, we study mathematical objects called vector spaces. A vector space is a set with a linear structure, as we shall define it in Section 2.2. We will try to understand these objects and the maps between them, look at ways to construct new vector spaces, and eventually classify all of them.