A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied (scaled) by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields instead. The operations of vector addition and scalar multiplication have to satisfy certain requirements, called axioms, listed below. An example of a vector space is that of Euclidean vectors which are often used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real factor is another force vector. The concrete nature of these operations depends on the type of vector under consideration, and can often be described by different means, e.g. geometric or algebraic. In view of the algebraic ideas behind these concepts explained below, the two operations are called vector addition and scalar multiplication.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.