Basis and Dimension

A basis is a minimal set of “building blocks” that can construct every vector in a space. The dimension tells you how many building blocks you need. These two ideas are central to everything that follows in linear algebra.

A basis for a vector space $V$ is a set of vectors that is:

1. Linearly independent (no redundancy)
2. Spans the entire space $V$ (every vector in $V$ can be written as a linear combination of the basis vectors)

The standard basis in 2D is $\{\langle 1, 0 \rangle, \langle 0, 1 \rangle\}$, often written as $\{\mathbf{i}, \mathbf{j}\}$. In 3D it’s $\{\mathbf{i}, \mathbf{j}, \mathbf{k}\}$.

The dimension of a vector space is the number of vectors in any basis for that space.

• $\dim(\mathbb{R}^2) = 2$
• $\dim(\mathbb{R}^3) = 3$
• The space of all $2 \times 2$ matrices has dimension 4

Key fact: all bases for the same vector space have the same number of vectors.

The span of a set of vectors is all possible linear combinations of those vectors. If a set of vectors spans the space and is linearly independent, it’s a basis.

In the diagram: the basis vectors $\mathbf{e}_1 = \langle 1, 0 \rangle$ (blue) and $\mathbf{e}_2 = \langle 0, 1 \rangle$ (green) are used to build the vector $\langle 3, 2 \rangle$ (purple). You scale $\mathbf{e}_1$ by 3 and $\mathbf{e}_2$ by 2, then add them. Any point in the plane can be reached this way, which is why these two vectors form a basis for $\mathbb{R}^2$.

Example 1: Standard Basis

The set $\{\langle 1, 0 \rangle, \langle 0, 1 \rangle\}$ is a basis for $\mathbb{R}^2$ because:

• They are linearly independent
• Any vector $\langle x, y \rangle = x\langle 1, 0 \rangle + y\langle 0, 1 \rangle$ (they span $\mathbb{R}^2$)

Example 2: Non-Standard Basis

Is $\{\langle 2, 0 \rangle, \langle 0, 3 \rangle\}$ a basis for $\mathbb{R}^2$?

Yes. They’re independent (neither is a scalar multiple of the other) and they span the space (you can reach any $\langle x, y \rangle$ as $\frac{x}{2}\langle 2, 0 \rangle + \frac{y}{3}\langle 0, 3 \rangle$).

Example 3: Not a Basis

The set $\{\langle 1, 1 \rangle, \langle 2, 2 \rangle\}$ is not a basis because the vectors are linearly dependent (the second is 2 times the first). They only span a single line, not all of $\mathbb{R}^2$.

Example 4: Dimension of a Plane in 3D

Any plane through the origin in $\mathbb{R}^3$ has dimension 2. You need exactly two linearly independent vectors to form a basis for that plane, even though the plane lives inside 3D space.

Basis and dimension show up everywhere:

• Game development: choosing efficient coordinate systems, compressing animation data, working with tangent spaces for normal mapping
• Computer graphics: changing between coordinate systems (world space, object space, camera space) is a change of basis
• Machine learning: PCA (Principal Component Analysis) finds the most important “directions” (basis) that explain the data
• Data compression: reducing dimension means storing less data while keeping the important information

Example: When a game stores character animations, it often uses a reduced basis (fewer independent directions) to save memory while still reconstructing realistic movement. This is dimension reduction in action.