Part III: Linear Algebra

In essence, linear algebra allows us to extend the concept of dimension beyond the usual 1-, 2- and 3-dimensional spaces. More specifically, we define the dimension of an equation to be the number of independent variables it contains, For example if f(x₁,x₂,x₃,x₄) and x₁,x₂,x₃ and x₄ are independent variables then the domain of f is 4-dimensional even though geometrically speaking there is no 4th dimension. So rather than talk about y = f(x) we, instead, "invent" a new variable (x₁,x₂,x₃,x₄) which we denote by a symbol such as X. We refer to (x₁,x₂,x₃,x₄) as a 4-tuple; and more generally, we refer to (x₁,x₂,x₃,x₄, ...x_n) as an n-tuple. In this context linear algebra is a study of the arithmetic of n-tuples.