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Subspaces

Module by: Doug Daniels, Steven Cox

Summary: 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.
Definition 1: A subset SS of a vector space VV is a subspace of VV when
1. if xx and yy belong to SS then so does x+y x y .
2. if xx belongs to SS and tt is real then tx t x belong to SS.
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 xx for which x 1 + x 2 + x 3 + x 4 =0 x 1 x 2 x 3 x 4 0 constitutes a subspace of ℝ 4 ℝ 4 . Can you 'see' this set? Do you 'see' that -1-100 -1-100 and -1010 -1010 and -1001 -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.
Definition 2: Span
A finite collection s 1 s 2 … s n s 1 s 2 … s n of vectors in the subspace SS is said to span SS if each element of SS can be written as a linear combination of these vectors. That is, if for each s∈S s S there exist nn reals x 1 x 2 … x n x 1 x 2 … x n such that s= x 1 s 1 + x 2 s 2 +…+ x n s n 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.
Definition 3: Linear Independence
A finite collection s 1 s 2 … s n s 1 s 2 … s n of vectors is said to be linearly independent when the only reals, x 1 x 2 … x n x 1 x 2 … x n for which x 1 + x 2 +…+ x n =0 x 1 x 2 … x n 0 are x 1 = x 2 =…= x n =0 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 s 1 s 2 … s n contains only the zero vector.
Combining these definitions, we arrive at the precise notion of a 'generating set.'
Definition 4: Basis
Any linearly independent spanning set of a subspace SS is called a basis of SS.
Though a subspace may have many bases they all have one thing in common:
Definition 5: Dimension
The dimension of a subspace is the number of elements in its basis.

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