In this chapter we have studied the plane and space as we know it from personal observation, and from our first geometry lessons at school. In addition to the well-known objects such as points, lines, and planes, the central theme was the concept of vector. We have learned how to calculate using vectors.

In particular, we have shown that coördinating planes and space happens systematically when we choose a basis of the plane or the space, and note the scalars required to write a vector as a linear combination of the basic vectors.

Before we got to work on angles between vectors, we introduced concepts such as dot product and cross product. We have also seen that, in order to reveal the given coordinate system of a basis, the basis must be orthonormal. Dot product and cross product are further of use when trying to find normal vectors, creating an equation for a plane, or finding the volume of a tetrahedron, prism, or parallelepiped, etc.

Along the way, we have seen many vector properties that can be used to calculate without geometric interpretation: we have gone from geometry to algebra, linear algebra in particular. The adjective 'linear' relates to the fact that almost every equation we encountered in this chapter was linear.

In the chapter Systems of linear equations and matrices, we will continue studying these equations. The chapter Vector spaces starts with the abstraction of geometry: the calculation rules for vectors that we learned in this chapter, are elevated to define an abstract space, which contains the same objects as points, lines, and planes. This approach is useful, not only for the fact that we will find higher-dimensional spaces, but also because these vector spaces occur in many mathematical systems. If you are already familiar with solving systems of linear equations, you can skip directly to this theory; if not, then you are advised to first read the section Systems of linear equations and matrices.