Vectors and Vector Spaces

A vector space over a field $F$ ( $\mathbb{R}$ or $\mathbb{C}$) is a set $V$ equipped With two operations:

1. Vector addition: $+ : V \times V \to V$
2. Scalar multiplication: $\cdot : F \times V \to V$

Satisfying the following axioms for all $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and all $\alpha, \beta \in F$:

1. Commutativity: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
2. Associativity of addition: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
3. Additive identity: There exists $\mathbf{0} \in V$ such that $\mathbf{v} + \mathbf{0} = \mathbf{v}$
4. Additive inverse: For each $\mathbf{v}$There exists $-\mathbf{v}$ such that $\mathbf{v} + (-\mathbf{v}) = \mathbf{0}$
5. Compatibility of scalar multiplication: $\alpha(\beta \mathbf{v}) = (\alpha\beta)\mathbf{v}$
6. Identity element of scalar multiplication: $1 \cdot \mathbf{v} = \mathbf{v}$
7. Distributivity over vector addition: $\alpha(\mathbf{u} + \mathbf{v}) = \alpha\mathbf{u} + \alpha\mathbf{v}$
8. Distributivity over scalar addition: $(\alpha + \beta)\mathbf{v} = \alpha\mathbf{v} + \beta\mathbf{v}$

Intuition. The abstract definition captures the algebraic structure shared by diverse objects: Geometric arrows, polynomials, functions, matrices. The axioms encode exactly what we need for Linear combinations to behave reasonably.

Example 1. $\mathbb{R}^n$ with component-wise addition and scalar multiplication is a vector space Over $\mathbb{R}$.

Example 2. The set $\mathcal{P}_n$ of all polynomials of degree at most $n$ with real coefficients, With the usual polynomial addition and scalar multiplication, is a vector space over $\mathbb{R}$. Its dimension is $n + 1$With standard basis $\{1, x, x^2, \ldots, x^n\}$.

Example 3. The set $C[a,b]$ of all continuous real-valued functions on $[a,b]$With point-wise Addition and scalar multiplication, is a vector space over $\mathbb{R}$. This space is Infinite-dimensional.

Example 4. The set $\mathcal{M}_{m \times n}(\mathbb{R})$ of all $m \times n$ real matrices is a Vector space over $\mathbb{R}$.

Example 5 (Function spaces). The set $\mathcal{F}(\mathbb{R}, \mathbb{R})$ of all functions $f : \mathbb{R} \to \mathbb{R}$ is a vector space over $\mathbb{R}$ under point-wise addition $(f + g)(x) = f(x) + g(x)$ and scalar multiplication $(\alpha f)(x) = \alpha \cdot f(x)$. The spaces $C^k(\mathbb{R})$ of $k$-times continuously differentiable functions and $L^2[a,b]$ of square-integrable functions are important subspaces of $\mathcal{F}(\mathbb{R}, \mathbb{R})$.

Example 6 (Sequence spaces). The set $\ell^2$ of all real sequences $(a_1, a_2, a_3, \ldots)$ With $\sum_{n=1}^{\infty} a_n^2 \lt \infty$ is a vector space over $\mathbb{R}$. This is the Infinite-dimensional analogue of $\mathbb{R}^n$ and is fundamental in functional analysis.

A subspace $W$ of a vector space $V$ is a subset $W \subseteq V$ that is itself a vector space Under the same operations.

Theorem 1.1 (Subspace Criterion). A non-empty subset $W \subseteq V$ is a subspace if and only If for all $\mathbf{u}, \mathbf{v} \in W$ and all $\alpha \in F$:

1. $\mathbf{u} + \mathbf{v} \in W$ (closed under addition)
2. $\alpha \mathbf{u} \in W$ (closed under scalar multiplication)

Proof. If $W$ is a subspace, closure is immediate from the definition. Conversely, if $W$ is Non-empty and closed under both operations, pick $\mathbf{u} \in W$. Then $-\mathbf{u} = (-1)\mathbf{u} \in W$ By closure under scalar multiplication, and $\mathbf{u} + (-\mathbf{u}) = \mathbf{0} \in W$ by closure Under addition. The remaining axioms are inherited from $V$. $\blacksquare$

Proposition 1.2 (Closure under Linear Combinations). If $W$ is a subspace of $V$Then $W$ is Closed under all finite linear combinations: for all $\mathbf{v}_1, \ldots, \mathbf{v}_k \in W$ and All $\alpha_1, \ldots, \alpha_k \in F$

$\alpha_1 \mathbf{v}_1 + \alpha_2 \mathbf{v}_2 + \cdots + \alpha_k \mathbf{v}_k \in W$

Proof. We proceed by induction on $k$. For $k = 1$, $\alpha_1 \mathbf{v}_1 \in W$ by closure under Scalar multiplication. Assume the result holds for $k - 1$ vectors. Then

$\alpha_1 \mathbf{v}_1 + \cdots + \alpha_k \mathbf{v}_k = (\alpha_1 \mathbf{v}_1 + \cdots + \alpha_{k-1} \mathbf{v}_{k-1}) + \alpha_k \mathbf{v}_k$

By the inductive hypothesis, $\alpha_1 \mathbf{v}_1 + \cdots + \alpha_{k-1} \mathbf{v}_{k-1} \in W$And $\alpha_k \mathbf{v}_k \in W$ by closure under scalar multiplication. Their sum is in $W$ by Closure under addition. $\blacksquare$

Example 7. The set of all solutions to the homogeneous equation $A\mathbf{x} = \mathbf{0}$ forms a Subspace of $\mathbb{R}^n$Called the null space of $A$.

Problem. Determine whether each of the following subsets of $\mathbb{R}^3$ is a subspace.

(a) $W_1 = \{(x, y, z) \in \mathbb{R}^3 : x + 2y - z = 0\}$

(b) $W_2 = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 = 1\}$

(c) $W_3 = \{(x, y, z) \in \mathbb{R}^3 : x = 0 \mathrm{~and~} y = z\}$

Problem. Let $U = \{(x, y, z) \in \mathbb{R}^3 : z = 0\}$ (the $xy$-plane) and $W = \{(x, y, z) \in \mathbb{R}^3 : x = 0\}$ (the $yz$-plane). Find $U + W$ and $U \cap W$ And verify the dimension formula.

• The empty set is not a vector space. The subspace criterion requires the subset to be non-empty. The trivial subspace $\{\mathbf{0}\}$ is the smallest subspace of any vector space.
• Non-homogeneous conditions do not define subspaces. The set of solutions to $A\mathbf{x} = \mathbf{b}$ with $\mathbf{b} \neq \mathbf{0}$ is not a subspace (it is an affine subspace, or coset of the null space).
• Closure must hold for all scalars. A set that is closed under addition and multiplication by positive scalars is not necessarily a subspace; it must also be closed under multiplication by $-1$.