Vector Spaces and Subspaces

What is a vector space?

A vector space over a field $\mathbb{F}$ (typically $\mathbb{R}$ or $\mathbb{C}$) is a set $V$ with two operations — addition and scalar multiplication — satisfying ten axioms. For every $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and $a, b \in \mathbb{F}$:

1. $\mathbf{u} + \mathbf{v} \in V$ (closure under addition)
2. $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ (commutativity)
3. $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$ (associativity)
4. There exists $\mathbf{0} \in V$ with $\mathbf{v} + \mathbf{0} = \mathbf{v}$ (zero vector)
5. Every $\mathbf{v}$ has an inverse $-\mathbf{v}$ with $\mathbf{v} + (-\mathbf{v}) = \mathbf{0}$
6. $a \mathbf{v} \in V$ (closure under scalar multiplication)
7. $a(b \mathbf{v}) = (ab) \mathbf{v}$
8. $1 \cdot \mathbf{v} = \mathbf{v}$
9. $a(\mathbf{u} + \mathbf{v}) = a \mathbf{u} + a \mathbf{v}$
10. $(a + b) \mathbf{v} = a \mathbf{v} + b \mathbf{v}$

Subspaces — the efficient test

A subspace of a vector space $V$ is a subset $W \subseteq V$ that is itself a vector space. To check whether $W$ is a subspace, you only need three things:

1. $W$ contains the zero vector $\mathbf{0}$.
2. $W$ is closed under addition: $\mathbf{u}, \mathbf{v} \in W \implies \mathbf{u} + \mathbf{v} \in W$.
3. $W$ is closed under scalar multiplication: $\mathbf{v} \in W, c \in \mathbb{F} \implies c \mathbf{v} \in W$.

The other seven axioms are inherited automatically from $V$.

Step-by-step — is the set of all 2D vectors a vector space?

$V = \mathbb{R}^2$, the set of all pairs $(x, y)$.

Zero vector: $\mathbf{0} = (0, 0) \in \mathbb{R}^2$ ✓

Closure under addition: if $(x_1, y_1)$ and $(x_2, y_2)$ are in $\mathbb{R}^2$, then $(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2) \in \mathbb{R}^2 ✓$

Closure under scalar multiplication: if $c \in \mathbb{R}$ and $(x, y) \in \mathbb{R}^2$: $c(x, y) = (cx, cy) \in \mathbb{R}^2 ✓$

The other axioms (commutativity, associativity, distributivity) follow from the same properties of real numbers. So $\mathbb{R}^2$ is a vector space.

Examples of subspaces of $\mathbb{R}^2$

• The zero subspace $\{\mathbf{0}\}$ — trivial but valid.
• Any line through the origin, e.g. $\{(t, 2t) : t \in \mathbb{R}\}$.
• $\mathbb{R}^2$ itself (every vector space is a subspace of itself).

Non-subspaces (common traps):

• A line not through the origin, e.g. $\{(x, y) : y = 2x + 1\}$. Fails the zero-vector test.
• The first quadrant $\{(x, y) : x \ge 0, y \ge 0\}$. Not closed under scalar multiplication (multiplying by $-1$ leaves the set).
• The unit disk $\{(x, y) : x^2 + y^2 \le 1\}$. Not closed under scalar multiplication ($2 \cdot (0.7, 0) = (1.4, 0)$ leaves the disk).

Why vector spaces matter

Vector spaces are the abstract setting where linear algebra lives. Once you have a vector space, concepts like linear combinations, independence, bases, and dimension all make sense — and they apply to far more than arrows in $\mathbb{R}^n$: polynomials, matrices, continuous functions, solutions to homogeneous ODEs, and more are all vector spaces.

Common mistakes

• Checking only one of the three subspace criteria. You need all three (plus the zero vector).
• Forgetting negative scalars. A set closed under positive scalars but not negative ones (like the first quadrant) fails.
• Assuming every line is a subspace. Only lines through the origin are subspaces; shifted lines are called affine subspaces, which are different.

Try it in the visualization

Draw candidate sets (lines through/off origin, half-planes, disks). Toggle the three subspace checks — each lights green or red based on whether the candidate satisfies it.