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3.8: Supplements - Subspaces

Last updated

Sep 17, 2022
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- 3.7: Supplements - Vector Space
- 3.9: Supplements - Row Reduced Form

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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: Subspace

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.

Definition: Span

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.

Definition: Linear Independence

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.'

Definition: 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:

Definition: Dimension

The dimension of a subspace is the number of elements in its basis.