Introduction: The Minimal Complete Set
A basis is a set of vectors that is both linearly independent and spans the entire space. It's the minimal set of "rulers" you need to describe any point in the space. Once you choose a basis, every vector gets a unique set of coordinates.
Formal Definition
A set \(\mathcal{B} = \{\vec{e}_1, \vec{e}_2, \ldots, \vec{e}_n\}\) is a basis for vector space \(V\) if:
- \(\mathcal{B}\) is linearly independent, and
- \(\text{span}(\mathcal{B}) = V\)
The number of vectors in a basis is called the dimension of the space: \(\dim(V) = n\).
The Standard Basis
In \(\mathbb{R}^n\), the standard basis consists of vectors with a single 1 and the rest 0s:
\[\vec{e}_1 = (1, 0, 0, \ldots), \quad \vec{e}_2 = (0, 1, 0, \ldots), \quad \ldots\]These are the "natural" coordinate axes we're used to.
Worked Examples
Example 1: Coordinates in the Standard Basis
Express \(\vec{v} = (3, 5)\) in the standard basis of \(\mathbb{R}^2\):
\[\vec{v} = 3\vec{e}_1 + 5\vec{e}_2 = 3(1, 0) + 5(0, 1)\]The coordinates are simply \((3, 5)\)—the numbers we started with!
Example 2: A Non-Standard Basis
Let \(\vec{b}_1 = (1, 1)\) and \(\vec{b}_2 = (1, -1)\). These are linearly independent and span \(\mathbb{R}^2\), so they form a basis.
Express \(\vec{v} = (3, 1)\) in this basis:
We need \(c_1(1, 1) + c_2(1, -1) = (3, 1)\)
- \(c_1 + c_2 = 3\)
- \(c_1 - c_2 = 1\)
Adding: \(2c_1 = 4 \Rightarrow c_1 = 2\). Then \(c_2 = 1\).
Result: In basis \(\{\vec{b}_1, \vec{b}_2\}\), the coordinates of \(\vec{v}\) are \((2, 1)\).
Example 3: Polynomial Basis
The space of polynomials of degree ≤ 2 has basis \(\{1, x, x^2\}\). Any polynomial \(p(x) = a + bx + cx^2\) has coordinates \((a, b, c)\) in this basis.
The Quantum Connection
Choosing a basis in quantum mechanics means choosing what observable you're measuring. The energy basis (eigenstates of the Hamiltonian) lets you read off energy probabilities. The position basis gives you the wavefunction \(\psi(x)\). The momentum basis gives you \(\tilde{\psi}(p)\). The physics is the same; only your "coordinate system" for describing it changes.