Subspaces 2.5 2001/08/09 2002/08/02 00:00:00.005 GMT-5 Doug Daniels rainking@alumni.rice.edu Steven Cox cox@rice.edu Doug Daniels rainking@alumni.rice.edu Mark Barrett markb@alumni.rice.edu Steven Cox cox@rice.edu subspace span linear independence basis dimension gauss gaussian gaussian elimination matrix methods matrix operations matrix reduction This module defines subspaces, span, linear independence, bases and dimension.

Subspace A subspace is a subset of a vector space that is itself a vector space. The simplest example is a line through the origin in the plane. For the line is definitely a subset and if we add any two vectors on the line we remain on the line and if we multiply any vector on the line by a scalar we remain on the line. The same could be said for a line or plane through the origin in 3 space. As we shall be travelling in spaces with many many dimensions it pays to have a general definition. A subset S of a vector space V is a subspace of V when if x and y belong to S then so does x y . if x belongs to S and t is real then t x belong to S. As these are oftentimes unwieldy objects it pays to look for a handful of vectors from which the entire subset may be generated. For example, the set of x for which x 1 x 2 x 3 x 4 0 constitutes a subspace of ℝ 4 . Can you 'see' this set? Do you 'see' that -1-100 and -1010 and -1001 not only belong to a set but in fact generate all possible elements? More precisely, we say that these vectors span the subspace of all possible solutions. Span A finite collection s 1 s 2 … s n of vectors in the subspace S is said to span S if each element of S can be written as a linear combination of these vectors. That is, if for each s S there exist n reals x 1 x 2 … x n such that s x 1 s 1 x 2 s 2 … x n s n . When attempting to generate a subspace as the span of a handful of vectors it is natural to ask what is the fewest number possible. The notion of linear independence helps us clarify this issue. Linear Independence A finite collection s 1 s 2 … s n of vectors is said to be linearly independent when the only reals, x 1 x 2 … x n for which x 1 x 2 … x n 0 are x 1 x 2 … x n 0 . In other words, when the null space of the matrix whose columns are s 1 s 2 … s n contains only the zero vector. Combining these definitions, we arrive at the precise notion of a 'generating set.' Basis Any linearly independent spanning set of a subspace S is called a basis of S. Though a subspace may have many bases they all have one thing in common: Dimension The dimension of a subspace is the number of elements in its basis.