Vector Space Axioms

• A vector space $V$ is a nonempty set of objects, called vectors.
• $V$ has two defined operations: addition and scalar multiplication.
• These operations must satisfy ten specific axioms to qualify $V$ as a vector space.

Vector Space Axioms

• Let $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ be vectors in $V$, and $c$ and $d$ be scalars.
• The sum $\mathbf{u} + \mathbf{v}$ is in $V$.
• $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$.
• $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$.
• $V$ contains a zero vector $\mathbf{0}$ such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$.
• For each $\mathbf{u}$ in $V$, there is a vector $-\mathbf{u}$ in $V$ where $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$.
• The scalar product $c\mathbf{u}$ is in $V$.
• $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$.
• $(c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}$.
• $c(d\mathbf{u}) = (cd)\mathbf{u}$.
• $1\mathbf{u} = \mathbf{u}$.
• A vector space is closed under addition and scalar multiplication.
• Vector spaces are more general than $\mathbb{R}^n$.

Vector Space Examples

• $\mathbb{R}^n$ represents vectors with $n$ entries.
• $\mathbb{P}_n$ is polynomials of degree at most $n$, such as $\mathbf{p}(t) = a_0 + a_1t + \dots + a_nt^n$.
• $M_{m \times n}$ signifies $m \times n$ matrices.
• The set of all real-valued functions defined on $\mathbb{R}$.

Subspaces of Vector Spaces

• A subspace $H$ of $V$ is a subset of $V$ that meets three conditions.
• A subspace is a vector space in its own right.
• The zero vector $\mathbf{0}$ is in $H$.
• For $\mathbf{u}$ and $\mathbf{v}$ in $H$, the sum $\mathbf{u} + \mathbf{v}$ is in $H$.
• For $\mathbf{u}$ in $H$ and scalar $c$, the scalar product $c\mathbf{u}$ is in $H$.
• $H$ must contain the zero vector.
• $H$ is required to be closed under both addition and scalar multiplication.

Subspace Examples

• Zero Subspace: The set containing only the zero vector in $V$ is a subspace of $V$.
• $V$ itself is a subspace of $V$.
• If $\mathbf{v}_1, \dots, \mathbf{v}_p$ exist in $V$, then $\text{Span} {\mathbf{v}_1, \dots, \mathbf{v}_p}$ is a subspace of $V$.

Basis of a Subspace

• For a subspace $H$ of $V$, a set ${\mathbf{v}_1, \dots, \mathbf{v}_p}$ in $V$ forms a basis for $H$ if:
• ${\mathbf{v}_1, \dots, \mathbf{v}_p}$ are linearly independent.
• $H = \text{Span} {\mathbf{v}_1, \dots, \mathbf{v}_p}$.

Basis Examples

• The standard basis for $\mathbb{R}^n$ serves as a basis for $\mathbb{R}^n$.
• The standard basis for $\mathbb{P}_n$ is ${1, t, \dots, t^n}$.
• The standard basis for $M_{2 \times 2}$ consists of the matrices: $$ \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix} $$