Linear Subspaces#

Definition: (Linear) Subspace

Let \((V,F,+,\cdot)\) be a vector space.

We say that \(U \subseteq V\) is a (linear) subspace of \((V,F,+,\cdot)\) if the following conditions hold simultaneously:

\(U \ne \varnothing\);
\(\mathbf{u}+\mathbf{u}' \in U\) for all \(\mathbf{u}, \mathbf{u}' \in U\);
\(c\mathbf{u}\in U\) for all \(\mathbf{u} \in U\) and all \(c \in F\).

Example

If \(\mathbf{v} \in V\), then the set

\[ U_{\mathbf{v}} \overset{\text{def}}{=} \{\lambda \mathbf{v} \mid \lambda \in F\} \]

is a subspace of \(V\).

TODO

Theorem: Alternative Definition

A subset \(U \subseteq V\) is a subspace of \(V\) if and only if it is equal to its own span:

\[ U = \mathop{\operatorname{span}}(U) \]

Proof

TODO

Theorem: Non-Subspace Criterion

Let \((V,F,+,\cdot)\) be a vector space and let \(U \subseteq V\).

If \(U\) does not contain the zero vector, then it is not a subspace.

Proof

TODO

Theorem: Subspace Intersection is a Subspace

Let \((V,F,+,\cdot)\) be a vector space.

If \(\mathcal{U}\) is a collection of subspaces of \(V\), then its intersection \(\bigcap \mathcal{U}\) is also a subspace of \(V\).

Proof

TODO

Theorem: Union of Subspaces

Let \((V,F,+,\cdot)\) be a vector space and let \(\mathcal{U}\) be a collection of subspaces of \(V\).

The union \(\bigcup \mathcal{U}\) is a subspace of \(V\) if and only if \(U_i \subseteq U_j\) or \(U_j \subseteq U_i\) for all \(U_i, U_j \in \mathcal{U}\).

Proof

TODO

Theorem: Subspace Dimension

If \(U\) is a subspace of finite dimensional vector space \(V\), then \(\dim U \le \dim V\) with \(\dim U = \dim V\) if and only if \(U = V\).

Definition: Codimension

The difference \(\dim V - \dim U\) is known as the codimension of \(U\).

Notation

\[ \mathop{\operatorname{codim}} U \]

Proof

TODO

Sum#

Definition: Sum

Let \(U_1, \dotsc, U_n\) be subspaces of a vector space \((V, F, +, \cdot)\).

The sum of \(U_1, \dotsc, U_n\) is the following set:

\[ U_1 + \cdots U_n \overset{\text{def}}{=} \{\mathbf{u}_1 + \cdots + \mathbf{u}_n \mid \mathbf{u}_1 \in U_1, \dotsc, \mathbf{u}_n \in U_n\} \]

Theorem: Sum is Subspace

If \(U_1, \dotsc, U_n\) are subspaces of a vector space \(V\), then their sum \(U_1 + \cdots + U_n\) is also a subspace \(V\).

Proof

TODO

Definition: Direct Sum

Let \(U_1, \dotsc, U_n\) be subspaces of a vector space \((V, F, +, \cdot)\).

We say that \(V\) is the direct sum of \(U_1, \dotsc, U_n\) if the following conditions are simultaneously true:

\(U_1 \cap \cdots \cap U_n = \{\mathbf{0}\}\).
For each \(\mathbf{v} \in V\), there exist \(\mathbf{u}_1 \in U_1, \dotsc, \mathbf{u}_n \in U_n\) such that \(\mathbf{v} = \mathbf{u}_1 + \cdots + \mathbf{u}_n\).

Notation

\[ V = U_1 \oplus \cdots \oplus U_n \]

Theorem: Uniqueness of Direct Sum Representation

If a vector space \((V, F, +, \cdot)\) is a direct sum of the subspaces \((U_1, F, +, \cdot), \dotsc, (U_n, F, +, \cdot)\), then the direct sum representation of each \(\mathbf{v} \in V\) is unique.

Proof

TODO