Basis and Dimension

Subtopic: Vector Spaces

A basis is a minimal spanning set—a collection of vectors that spans the space without any redundancy. The dimension of a vector space is the number of vectors in any basis. These concepts let us measure the "size" of infinite sets in a meaningful way: (R^3) is 3-dimensional because you need exactly 3 independent vectors to span it.

Introduction

How big is (R^3)? It contains infinitely many vectors, but in some sense it's smaller than (R^4). The notion of dimension captures this. You need three independent directions to reach any point in (R^3), but four for (R^4).

A basis gives coordinates: once you fix a basis, every vector has a unique address—its coordinates relative to that basis. Different bases give different coordinate systems, but the number of coordinates (the dimension) stays the same.

Definition

Basis

A basis for a vector space V is a set of vectors B={(v_1),(v_2),…,(v_n)} that is:

1. Linearly independent: No vector is a combination of the others

2. Spanning: Every vector in V is a combination of the basis vectors

Equivalently, a basis is a maximal independent set, or a minimal spanning set.

Dimension

The dimension of a vector space V, denoted dim*(V), is the number of vectors in any basis of V.

A remarkable theorem guarantees this is well-defined: all bases of a finite-dimensional space have the same number of elements.

Geometric Interpretation

Dimension counts the degrees of freedom needed to specify a point. A line is 1-dimensional (one number suffices), a plane is 2-dimensional, and ordinary space is 3-dimensional.

A basis provides a coordinate system. In (R^2), the standard basis {(1,0),(0,1)} gives Cartesian coordinates. The basis {(1,1),(1,−1)} gives a different coordinate system—same plane, different addressing scheme.

Standard Bases

For (R^n), the standard basis is:

(e_1)=(1,0,…,0),(e_2)=(0,1,…,0),…,(e_n)=(0,0,…,1)

Any vector ((x_1),(x_2),…,(x_n))=(x_1)*(e_1)+(x_2)*(e_2)+⋯+(x_n)*(e_n). The coordinates are just the components.

For the polynomial space (P_n) (polynomials of degree ≤n), a standard basis is {1,x,x2,…,xn}. Dimension: n+1.

For the matrix space (M_m×n), a standard basis consists of matrices with a single 1 and zeros elsewhere. Dimension: m*n.

Finding a Basis

From a Spanning Set

If S spans V but isn't independent, remove redundant vectors:

1. Put vectors as columns of a matrix

2. Row reduce to find pivot columns

3. The original vectors corresponding to pivot columns form a basis

From an Independent Set

If S is independent but doesn't span V, extend it:

1. Find a vector not in span (S)

2. Add it to S (the new set is still independent)

3. Repeat until you span V

Worked Example

Find a basis for the subspace W={(x,y,z)∈(R^3):x+y−z=0}.

Solve for one variable: z=x+y. Vectors in W have the form:

(x,y,x+y)=x(1,0,1)+y(0,1,1)

So W=span{(1,0,1),(0,1,1)} .

Are these independent? Set (c_1)(1,0,1)+(c_2)(0,1,1)=(0,0,0):

(c_1)=0, (c_2)=0, (c_1)+(c_2)=0 — only solution is (c_1)=(c_2)=0. ✓ Independent.

Basis: {(1,0,1),(0,1,1)}. Dimension: 2.

W is a plane (2D subspace of (R^3)).

Key Theorems

All Bases Have the Same Size

If V has a basis of n vectors, then every basis of V has exactly n vectors. This makes dimension well-defined.

Dimension Bounds

In an n-dimensional space:

• Any linearly independent set has at most n vectors

• Any spanning set has at least n vectors

• Any independent set of n vectors is automatically a basis

• Any spanning set of n vectors is automatically a basis

Subspace Dimension

If W is a subspace of V, Then dim(W)≤dim(V), with equality if and only if W=V.

Coordinates

Given a basis B={(v_1),…,(v_n)}, every vector v∈V has a unique representation:

v=(c_1)*(v_1)+(c_2)*(v_2)+⋯+(c_n)*(v_n)

The coefficients ((c_1),(c_2),…,(c_n)) are the coordinates of v relative to basis B, written [v]B.

Example: In (R^2) with basis B={(1,1),(1,−1)}, the vector (3,1) has coordinates [v]B=(2,1) because (3,1)=2*(1,1)+1*(1,−1).

Applications

In computer graphics, choosing a basis aligned with an object simplifies transformations. Local coordinates make rotation easier.

In data science, PCA finds an optimal basis where the first few coordinates capture most of the variance, enabling dimensionality reduction.

In quantum mechanics, measuring an observable corresponds to projecting onto its eigenbasis. Different bases reveal different properties.

In signal processing, different bases (Fourier, wavelet) reveal different signal features.

Summary

A basis is a linearly independent spanning set—no redundancy, full coverage. The dimension is the size of any basis (all bases have the same size). Standard bases provide natural coordinates; other bases can simplify specific problems. To find a basis for a subspace, parameterize the subspace and read off spanning vectors, then verify independence. The dimension of a subspace tells you how many free parameters determine it.