Standard Basis

The standard basis of Rn\mathbb{R}^n (also called the canonical basis or natural basis) is the default coordinate system against which every other vector in Rn\mathbb{R}^n is decomposed. It consists of nn vectors, each pointing purely along one axis.

Definition

The standard basis of Rn\mathbb{R}^n is the set {e1,…,en}\{\mathbf{e}_1, \ldots, \mathbf{e}_n\}, where each unit coordinate vector (also known as a standard unit vector or coordinate unit vector) ei\mathbf{e}_i has a 11 in position ii and 00 everywhere else:

ei=(0,…,0,1⏟i-th slot,0,…,0)⊤,i∈{1,…,n}\mathbf{e}_i = (0, \ldots, 0, \underbrace{1}_{i\text{-th slot}}, 0, \ldots, 0)^\top, \quad i \in \{1, \ldots, n\}

In R3\mathbb{R}^3:

e1=(100),e2=(010),e3=(001)\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \qquad \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \qquad \mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}

In physics and engineering, these three vectors are often written i^,j^,k^\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} (or x^,y^,z^\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}), where the hat denotes unit length.

Key Properties

For ei,ej\mathbf{e}_i, \mathbf{e}_j in the standard basis of Rn\mathbb{R}^n:

  • Unit length: ∥ei∥=1\|\mathbf{e}_i\| = 1
  • Mutual orthogonality: ⟨ei,ej⟩=δij\langle \mathbf{e}_i, \mathbf{e}_j \rangle = \delta_{ij}, where δij\delta_{ij} is the Kronecker delta (11 if i=ji = j, else 00)
  • Coordinate expansion: every x=(x1,…,xn)⊤∈Rn\mathbf{x} = (x_1, \ldots, x_n)^\top \in \mathbb{R}^n decomposes uniquely as
x=∑i=1nxiei,xi=⟨x,ei⟩\mathbf{x} = \sum_{i=1}^{n} x_i \mathbf{e}_i, \qquad x_i = \langle \mathbf{x}, \mathbf{e}_i \rangle

The last property is why the standard basis is the default reference frame: any vector is literally the sum of its coordinates times the basis vectors, and each coordinate is recovered by an inner product against the corresponding ei\mathbf{e}_i.