W between vector spaces V and W which preserve the vector space structure. There are two important subspaces attached to such a map. We will look again in particular at systems of linear equations. Linear maps can be represented by matrices and this connection will be discussed in detail, including the introduction of determinants, eigenvectors and eigenvalues. Much of this theory can be done for any field of scalars.
Vector spaces over the real numbers are special in that they allow measurements of length and angle.
There will be a short introduction to orthogonality and an algorithm which produces orthogonal vectors. Analysis is that part of mathematics concerned with functions and graphs. Throughout the 17th and 18th centuries, problems in mathematics and physics were being successfully solved using ideas that we now call Differential Equations, Fourier Analysis, and Fixed Point Theorems. Despite these successes, much of the work was not accurate and many basic questions were left unanswered. During the 19th century these problems were formulated precisely and solved.