3.8: Supplements - Subspaces
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 \(tx\) 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 \(\mathbb{R}^{4}\). Can you 'see' this set? Do you 'see' that
\[\begin{pmatrix} {-1}\\ {1}\\ {0}\\ {0} \end{pmatrix} \nonumber\]
and
\[\begin{pmatrix} {-1}\\ {0}\\ {1}\\ {0} \end{pmatrix} \nonumber\]
and
\[\begin{pmatrix} {-1}\\ {0}\\ {0}\\ {1} \end{pmatrix} \nonumber\]
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.
A finite collection \(\{s_{1}, s_{2}, \cdots, s_{n}\}\) 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 \(s \in S\) there exist nn reals \(\{x_{1}, x_{2}, \cdots, x_{n}\}\) such that \(s = x_{1}s_{1}+x_{2}s_{2}+ \cdots +x_{n}s_{n}\).
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.
A finite collection \(\{s_{1}, s_{2}, \cdots, s_{n}\}\) of vectors is said to be linearly independent when the only reals, \(\{x_{1}, x_{2}, \cdots, x_{n}\}\) for which \(x_{1}+x_{2} + \cdots+x_{n} = 0\) are \(x_{1} = x_{2} = \cdots = x_{n} = 0\) In other words, when the null space of the matrix whose columns are \(\{s_{1}, s_{2}, \cdots, s_{n}\}\) contains only the zero vector.
Combining these definitions, we arrive at the precise notion of a 'generating set.'
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:
The dimension of a subspace is the number of elements in its basis.