Finite Dimensional Vector Spaces
This chapter establishes many important results related to the vector spaces. It also introduces the general class of sets called “vector spaces” that can be defined as algebraic structures with operations having properties similar to those of addition and scalar multiplication on R n . It proves eight properties of addition and scalar multiplication in R n and M mn . These properties are important because all other results involving these operations can be derived from them. It also introduces the general class of vector spaces, having operations of addition and scalar multiplication with these same eight properties, as well as two closure properties. The two closure properties require that both the operations of vector addition and scalar multiplication always produce an element of the vector space as a result. Some vector spaces can have additional operations. For example, R n has the dot product and M nn has matrix multiplication and the transpose. In particular, there is no such operation as multiplication or division of one vector by another in general vector spaces. The only general vector space operation that combines two vectors is vector addition. By studying vector spaces whose objects share many of the same properties of vectors in R n, a more abstract theory with a wider range of applications is explored.