ABSTRACT
In this chapter the theoretical foundation for the concepts introduced in the earlier chapters is given. In Section 1.2, the addition and scalar multiplication of vectors was defined and in Section 1.3, the addition and scalar multiplication of matrices was introduced. In both the cases the addition and multiplication of real numbers has been imbibed naturally. Further, it may be noted that the addition and scalar multiplication of polynomials has been developed in a similar way. On observing the definitions of addition and scalar multiplication of the above fore - mentioned different setups, one can observe a similarity that is present and discover the cord that runs through all of them. This similarity helps in generalizing the concept and in taking the definitions to the next level which is abstraction. This is similar to the set up where children are first given chocolates to add and then are taught with the numbers and later on with symbols like a, b, c, etc.
Sections 3.1 to 3.3 of this chapter deal with abstract concepts of a vector space. Section 3.4 deals with the change of basis matrix, an essential tool to be learnt by students of computer science and robotics among others. Section 3.5 concentrates on linear transformations. As matrices are very useful in studying applied areas, linear transformations are studied using the matrix defined by them in Section 3.6. A generalization of the dot product or inner product in a vector space is introduced leading to an inner product space. Further, the notion of the inner product between two vectors, which is an abstraction of the concept of angle between two vectors in geometry, is utilized to define orthogonal and orthonormal sets and this is the content of Section 3.7. The singular value decomposition of a matrix is also dealt in this section. These topics lead to the presentation of the widely used Gram Schmidt orthogonalization process given in Section 3.8. Finally, an attempt is made to link linear algebra with differential equations through a couple of special functions in Section 3.9.
