Vector Spaces

Picture a vector space: large and expansive with no ends in sight. Subtle tick-marks line the ether. There are an arbitrary number of dimensions; any finite number n will do (infinite-dimensional vector spaces make up a different field of study).

Picture a basis in this vector space: a set of vectors which demarcate the space. Points in the vector space are described through their coordinates with respect to a particular basis. Linear algebra vastly generalizes the coordinate system: the “axes” (basis vectors) may be arbitrary in number (as the space is arbitrary in dimension) and they need not be perpendicular nor of equal lengths. These basis vectors simply must be (a) linearly independent (or non-redundant) and (b) span the whole space. Any single vector on a line, or two non-colinear vectors on a plane, or three non-coplanar vectors in three-space, are linearly independent – they can’t be expressed as combinations of each other – and they are a spanning set – the whole space can be accessed through their combinations. Dimension, as it turns out, describes nothing more than the maximum possible number of linearly independent vectors – how many unique vectors might one introduce? or, equivalently, the minimum number of spanning vectors – how many must we add to span the whole space? Such a “minimum spanning set” is called a basis. Can you imagine a fourth vector that can’t be described through some sum the other three, or a space which requires four vectors to span? Then you have imagined the fourth dimension.

Continue reading