Linear vector space is an important concept in linear algebra. Linear vector spaces have applications in different fields of engineering. In the discipline of electrical engineering, vector spaces are extensively used in detection and estimation theory, color engineering, signal and image processing and statistical communication theory. The concept of normed and inner product vector spaces is the foundation of optimization theory and is widely used in solving engineering problems such as optimal control and operations research type problems including inventory control. The rich theory of vector spaces is the foundation of modern engineering and computational algorithms used to solve various engineering problems. This chapter addresses the concept of linear vector spaces and normed and inner product spaces with applications in signal and image processing, communication and color engineering.