Dimensions and linear independence
T
he dimension of an object or space is a measure of its size. For standard Euclidian space it is the number of coordinates needed to specify the points within that space. For instance, a circle is onedimensional, a disc is two-dimensional, and a sphere is three-dimensional. Intuitively we understand that there are two or three directions that can be explored: up, down, and sideways. This is expressed mathematically using the idea of independence. A set of vectors is linearly independent if none of the vectors can be written as the sum of multiples of the others. Any set of n linearly independent vectors is said to be a basis for n-dimensional space, and any vector in the space can be written as a linear combination of basis vectors. In three dimensions, the standard Cartesian basis is the set of coordinate vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1), which have the additional property that they are perpendicular to each other. But any three linearly independent vectors is an acceptable basis for three dimensions.