An Intuitive Introduction to Linear Algebra
- Steven Lehar
If you had trouble learning linear algebra, it was because it was not taught in pictures! Linear algebra is a highly spatial concept, it becomes intuitively obvious when you "see the picture" in your mind. There are several "problems" with the conventional "Gibbs" matrix notation that we all learned in school. First, the matrix is squashed into a two-dimensional representation to fit the printed page, even for a 3-dimensional matrix that actually represents a three-dimensional structure. Second, the matrix is indexed in (column, row) or (y,x) order, which is the reverse of the more familiar (row, column) or (x,y) order of Cartesian coordinates. Third, the matrix index starts 1, with row = (1,2,3...) whereas the Cartesian coordinates are indexed from zero in (0, 1, 2...) sequence. Fourth, vectors in Gibbs vector notation are summarized to the (x,y,z) coordinates of their endpoint, which obscures the fact that they actually represent a vector from the origin in that direction. It seems almost like a deliberate attempt to obscure the beautiful spatial concepts hidden in the spatial transforms of linear algebra. This demo shows how the counter-intuitive array of numbers encoded in the classical matrix of linear algebra relate to the beautiful spatial concepts that they truly represent, to reveal how the basic ideas behind linear algebra are beautifully simple and intuitive, easy for anyone to understand and appreciate.