20. Vector Spaces and Subspaces

Dated: 23-04-2025

Definition

Let \(\mathbb V\) be an arbitrary non empty set¹ of objects on which 2 operations are defined, addition and multiplication by scalar numbers.
If following axioms are satisfied by all the objects \(u, v, w \in \mathbb V\) and all scalars \(k\) and \(l\) then \(\mathbb V\) is called a vector space.

Axioms of Vector Space

Closure Property

\[\vec u , \vec v \in \mathbb V \implies \vec u + \vec v \in \mathbb V\]

Commutative Property

\[\vec u , \vec v \in \mathbb V \implies \vec u + \vec v = \vec v + \vec u\]

Associative Property

\[\vec u , \vec v , \vec w \in \mathbb V \implies \vec u + (\vec v + \vec w) = (\vec u + \vec v) + \vec w\]

Additive Identity

\[\vec u \in \mathbb V \text{ and } \exists \, \vec 0 \in \mathbb V \implies \vec u + \vec 0 = \vec 0 + \vec u = \vec u\]

Additive Inverse

\[\vec u \in \mathbb V \text{ and } \exists \, \vec {- u} \in \mathbb V \implies \vec u + \vec {-u} = \vec {- u} + \vec u = \vec 0\]

Scalar Multiplication

\[\vec u \in \mathbb V \land k \in \mathbb R \implies k \vec u \in \mathbb V\]

Distributive Law

\[\vec u, \vec v \in \mathbb V \land k \in \mathbb R \implies k (\vec u + \vec v) = k \vec u + k \vec v\]

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\[\vec u \in \mathbb V \land m, n \in \mathbb R \implies (m + n) \vec u = m \vec u + n \vec u\]

\[\vec u \in \mathbb V \land m, n \in \mathbb R \implies m (n \vec u) = n(m \vec u)\]

\[\vec u \in \mathbb V \implies \vec 1 \vec u = \vec u\]

Definition

A subset¹ \(\mathbb W\) of a vector space \(\mathbb V\) is called a subspace of \(\mathbb V\) if \(\mathbb W\) itself is a vector space under the addition and scalar multiplication defined on \(\mathbb V\).

Since \(\mathbb W\) is a subspace of \(\mathbb V\), most of the axioms of \(\mathbb V\) are inherited to \(\mathbb W\) and thus, do not need verification.

Theorem

\(\mathbb W\) is called the subspace of \(\mathbb V\) only and only if \(\mathbb W\) is closed under addition and scalar multiplication.

Every vector space has at least 2 subspaces.

• Itself
• \(\{\vec 0\}\) called the zero subspace.

If \(\mathbb V\) is a subspace then not every subset¹ of it is a subspace.

Theorem

If \(\vec {v_1}, \vec {v_2}, \ldots, \vec {v_p} \in \mathbb V\) then \(Span\{\vec{v_1}, \vec{v_2}, \ldots, \vec{v_p}\}\) is a subspace of \(\mathbb V\).

References

Read more about notations and symbols.

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1. Read more about sets. ↩↩↩