Linear Subspaces
- Linear Algebra 1.2
Linear subspace
If $V$ is vector space over $F$ and $W$ is a subset of $V$, then $W$ is a linear subspace of $V$ if $W$ is a vector space over $F$ under the operations of $V$.
Equivalently, a nonempty subset $W$ is a subspace of $V$ if arbitrary linear combinations of some vectors of $W$ are also an element of $W$.
- $ \b{0} \in W \subseteq V $
- $ \b{w}_1,\b{w}_2\in W \implies \b{w}_1+\b{w}_2\in W$
- $ \b{w}\in W, c\in F \implies c\b{w}\in W$
- $ \b{w}\in W \implies -\b{w}\in W$
The last condition is redundant, since $-\b{w} = (-1)\b{w}$. It’s sometimes denoted as $W \le V$ when $W$ is a subspace of $V$, but these aren’t common enough to be used without explicitly specifying their meaning first. Especially, if $W$ is a proper subspace of $V$, we can write $W < V$.
As a corollary, all vector spaces are equipped with at least two linear subspaces:
- The zero subspace $\{\b{0}\}$
- The vector space itself
These are called the trivial subspaces of the vector space.
Properties
Let $\mathcal{C}$ be a collection of subspaces of $V$, and let $W = \bigcap_{U\in\mathcal{C}} U$ be the intersection of all subspaces in $\mathcal{C}$. It is trivial that $\b{0}\in W$ since $\b{0}$ is in every subspace of $V$. For $\forall U\in\mathcal{C}$, $\b{w}_1,\b{w}_2\in W \subseteq U$ implies $\b{w}_1+\b{w}_2\in U$, and $c\b{w}\in U$ for $\b{w}\in W \subseteq U$ and $c\in F$. Thus, $\b{w}_1+\b{w}_2\in W$ and $c\b{w}\in W$. Since $W$ satisfies the three conditions of being a subspace, $W$ is a subspace of $V$.
Examples
Transpose
The transpose of a matrix $A \in \mathcal{M}_{m\times n}(F)$ is the matrix $A^\top \in \mathcal{M}_{n\times m}(F)$ obtained by interchanging the rows and columns of $A$.
\[ \left( A^\top \right)_{ij} = A_{ji} \]
for $1 \le i \le n$ and $1 \le j \le m$. The following properties hold:
- $(aA + bB)^\top = aA^\top$
- $\left( A^\top \right)^\top = A$
Symmetric and skew-symmetric matrices
A symmetric matrix is a square matrix that is equal to its transpose, i.e., $A = A^\top$. A skew-symmetric matrix is a square matrix that satisfies $A^\top = -A$. The set of all symmetric and skew-symmetric $n \times n$ matrices over $F$ also forms a linear subspace of $\mathcal{M}_{n\times n}(F)$. We often denote them as $\mathrm{Sym}_n(F)$ and $\mathrm{Skew}_n(F)$, respectively.
- $A+A^\top$ is symmetric
- $A-A^\top$ is skew-symmetric
Triangular and diagonal matrices
A square matrix $A$ is called upper triangular if all entries below the main diagonal are zero, i.e., $A_{ij} = 0$ for $i > j$. It is called lower triangular if all entries above the main diagonal are zero, i.e., $A_{ij} = 0$ for $i < j$. The set of all upper or lower triangular $n \times n$ matrices over $F$ also forms a linear subspace of $\mathcal{M}_{n\times n}(F)$.
A square matrix $M$ is called diagonal matrix if all of its non-diagonal entries are zero, i.e., $M_{ij} = 0$ for $i \neq j$. The set of all diagonal $n \times n$ matrices over $F$ forms a linear subspace of $\mathcal{M}_{n\times n}(F)$. Diagonal matrices are symmetric, but not all symmetric matrices are diagonal.
Trace
The trace of a $n \times n$ square matrix $A$ is the sum of its diagonal entries, denoted as:
\[ \tr(A) = \sum_{i=1}^n A_{ii} \]
Trace is a linear functional on $\mathcal{M}_{n\times n}(F)$.
- $\tr(aA + bB) = a\tr(A) + b\tr(B)$
Direct sum
Given two nonempty subsets $S_1,S_2 \subset V$, the sum of $S_1$ and $S_2$ is defined as:
\[ S_1 + S_2 = \Set{ \b{v}_1 + \b{v}_2 | \b{v}_1 \in S_1, \b{v}_2 \in S_2 } \]
For the family of subsets $\mathcal{F} = \Set{ S_i | i \in I }$, the sum can be notated as:
\[ \sum \mathcal{F} = \sum_{i \in I} S_i \]
If $S_1$ and $S_2$ are linear subspaces of $V$ such that $S_1 \cap S_2 = \{\b{0}\}$ and $S_1 + S_2 = V$, then $V$ is called the direct sum of $S_1$ and $S_2$, denoted as:
\[ V = S_1 \oplus S_2 \]
For the family of subspaces $\mathcal{F} = \Set{ S_i | i \in I }$, the direct sum can be notated as:
\[ \bigoplus \mathcal{F} = \bigoplus_{i \in I} S_i \]
It is defined if and only if for every $i \in I$,
\[ S_i \cap \left( \sum_{j \in I, j \neq i} S_j \right) = \{\b{0}\} \]
This condition is equivalent to the following condition: for every $\b{v} \in \sum \mathcal{F}$,
- there exist unique $\b{s}_i \in S_i$ for $i \in I$ such that $\b{v} = \sum_{i \in I} \b{s}_i$
and it is also equivalent to the following condition:
- $\b{0} = \sum_{i \in I} \b{s}_i$ implies $\b{s}_i = \b{0}$ for every $i \in I$
Proof is omitted.
Complementary subspace
If $W$ is a linear subspace of $V$, then a complementary subspace of $W$ in $V$ is a linear subspace $U$ of $V$ such that:
- $V = W \oplus U$
For example, we have:
- $\mathcal{M}_n(F) = \mathrm{Sym}_n(F) \oplus \mathrm{Skew}_n(F)$
It can be shown that for every linear subspace $W$ of $V$, there exists a complementary subspace of $W$ in $V$ (though it may not be unique). Its proof requires the knowledge of the basis of a vector space, so it will be covered in the next section.
External direct sum
The direct sum defined above is occasionally called the internal direct sum to distinguish it from the external direct sum. Given vector spaces $V_1,\dots,V_n$ over $F$, the external direct sum of $V_1,\dots,V_n$ is the vector space defined as:
\[ V_1 \boxplus \cdots \boxplus V_n = \Set{ (\b{v}_1,\dots,\b{v}_n) | \b{v}_i \in V_i, 1 \le i \le n } \]
with componentwise addition and scalar multiplication. In common, for the family of vector spaces $\mathcal{F} = \Set{ V_i | i \in I }$, the external direct sum can be defined as:
\[ \bigoplus {}^\text{ext} \mathcal{F} = \bigoplus_{i \in I} {}^\text{ext} V_i = \Set{ f : I \to \bigcup_{i \in I} V_i | f(i) \in V_i, \abs{ \supp f } < \infty } \]
Direct product or cartesian product is a more general construction that doesn’t require the finiteness condition on the support of $f$.
\[ \prod \mathcal{F} = \prod_{i \in I} V_i = \Set{ f : I \to \bigcup_{i \in I} V_i | f(i) \in V_i } \]
For finite family of vector spaces, the external direct sum and the cartesian product are identical, but for infinite family of vector spaces, the external direct sum is a proper subspace of the cartesian product.
Coset and quotient space
Given a linear subspace $W$ of $V$ and a vector $\b{v} \in V$, the coset of $W$ containing $\b{v}$ is defined as:
\[ \{ \b{v} \} + W = \Set{ \b{v} + \b{w} | \b{w} \in W } \]
It is customary to denote this coset by $\b{v} + W$ instead of $\{ \b{v} \} + W$. Addition and scalar multiplication on the set of cosets $ S = \Set{ \b{v} + W | \b{v} \in V } $ can be defined properly, so that $S$ forms a vector space over $F$, called the quotient space of $V$ by $W$, denoted as $V/W$.
\[ V/W = \{ \b{v} + W | \b{v} \in V \} \]
If $V = W \oplus U$ for some linear subspace $U$ of $V$, then we can show that $V/W$ is isomorphic to $U$. We will cover the concept of isomorphism later, but for now, we can understand that $V/W$ and $U$ are “the same” vector space in the sense that they have the same structure as vector spaces.
Lattice of subspaces
The set $\mathcal{S}(V)$ of all linear subspaces of $V$ is partially ordered by set inclusion. The zero subspace is the least element of $\mathcal{S}(V)$, and $V$ itself is the greatest element of $\mathcal{S}(V)$. If we consider $(\mathcal{S}(V),\le)$ as a poset(partially ordered set), then we can show that for $S_1,S_2\in\mathcal{S}(V)$:
- $S_1 + S_2 = \sup(S_1,S_2)$
- $S_1 \cap S_2 = \inf(S_1,S_2)$
where the supremum and infimum are taken with respect to the partial order $\le$, and return the element of $\mathcal{S}(V)$. Thus, $(\mathcal{S}(V),+,\cap)$ is a lattice, where the join operation is the sum of subspaces and the meet operation is the intersection of subspaces.
