Linear algebra is undoubtedly one of the most powerful structures of pure and applied mathematics. It combines extreme generality with deeply held spatial intuitions. In economics and data science, one would dare to say, lies at the basis of most of the other mathematical techniques used. Yet, standard presentations of the subject tend to refrain from displaying the full extent of the deeply intuitive nature of the underlying structures, despite the fact that such intuitions are so useful when applying linear algebra, and when extending techniques to tackle nonlinear problems. In the context of the “Intuitive Mathematical Economics Series”, this is the first paper of a series dedicated to presenting linear algebra emphasizing both, its intuitive, and its general nature. In this case we present linear manifolds and vector spaces.