Bases and Coordinates
Motivation
A vector space is an abstract collection of objects that can be added and scaled. To compute with them, we need to turn each vector into a list of numbers — its coordinates. A basis is the apparatus that lets us do this: a small set of reference vectors so well-chosen that every vector in the space has exactly one expansion in terms of them (Axler 2015).
Bases turn abstract problems into concrete arithmetic. They let us represent linear maps as matrices, compare different ways of looking at the same data, change perspective (rotate, project, decorrelate) without losing information, and define dimension itself. Without bases, linear algebra would still be true, but it would not be usable.
Definition
A basis of a vector space \(V\) is a set of vectors \(\mathbf{b}_1, \ldots, \mathbf{b}_n \in V\) that is
- linearly independent — the only solution to \(c_1 \mathbf{b}_1 + \cdots + c_n \mathbf{b}_n = \mathbf{0}\) is \(c_i = 0\) for all \(i\), and
- spanning — every \(\mathbf{v} \in V\) can be written as some linear combination \(\mathbf{v} = c_1 \mathbf{b}_1 + \cdots + c_n \mathbf{b}_n\).
Independence and spanning together guarantee that every vector has a unique expansion: existence comes from spanning, uniqueness from independence.
The number \(n\) of vectors in a basis is the dimension \(\dim V\). Although a vector space has infinitely many bases, all of them have the same size — so dimension is a property of the space, not of the chosen basis.
Standard basis of \(\mathbb{R}^n\)
The vectors
\[ \mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \quad \mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix}, \quad \ldots, \quad \mathbf{e}_n = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix} \]
form the standard basis of \(\mathbb{R}^n\). The coordinates of a vector with respect to this basis are simply its entries.
Monomial basis of polynomials
The polynomials \(1, x, x^2, \ldots, x^d\) form a basis of \(P_d\) (polynomials of degree at most \(d\)). The coordinates of \(p(x) = a_0 + a_1 x + \cdots + a_d x^d\) in this basis are \((a_0, a_1, \ldots, a_d)\).
Coordinates
Given a basis \(\mathcal{B} = (\mathbf{b}_1, \ldots, \mathbf{b}_n)\) of \(V\) and a vector \(\mathbf{v} \in V\), the unique scalars \(c_1, \ldots, c_n\) satisfying
\[ \mathbf{v} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 + \cdots + c_n \mathbf{b}_n \]
are the coordinates of \(\mathbf{v}\) in the basis \(\mathcal{B}\), written
\[ [\mathbf{v}]_\mathcal{B} = \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{pmatrix} \in \mathbb{R}^n. \]
The map \(\mathbf{v} \mapsto [\mathbf{v}]_\mathcal{B}\) is a one-to-one correspondence between \(V\) and \(\mathbb{R}^n\). It turns every \(n\)-dimensional vector space into a working copy of \(\mathbb{R}^n\).
A non-standard basis of \(\mathbb{R}^2\)
Let
\[ \mathbf{b}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad \mathbf{b}_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}. \]
These are linearly independent and span \(\mathbb{R}^2\), so they are a basis. The vector \(\mathbf{v} = (3, 1)^\top\) in standard coordinates has coordinates in \(\mathcal{B}\) given by solving
\[ c_1 \begin{pmatrix} 1 \\ 1 \end{pmatrix} + c_2 \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \;\Longrightarrow\; c_1 = 2,\; c_2 = 1. \]
So \([\mathbf{v}]_\mathcal{B} = (2, 1)^\top\). The same point in the plane has standard coordinates \((3, 1)\) and \(\mathcal{B}\)-coordinates \((2, 1)\).
Change of Basis
If \(\mathcal{B} = (\mathbf{b}_1, \ldots, \mathbf{b}_n)\) is a basis of \(\mathbb{R}^n\), stack the basis vectors as columns of a matrix \(B = [\mathbf{b}_1 \;\cdots\; \mathbf{b}_n] \in \mathbb{R}^{n \times n}\).
Then the relation between coordinates and standard-coordinate vectors is
\[ \mathbf{v} = B \cdot [\mathbf{v}]_\mathcal{B}, \qquad [\mathbf{v}]_\mathcal{B} = B^{-1} \mathbf{v}. \]
\(B\) is called the change-of-basis matrix from \(\mathcal{B}\) to the standard basis. Since \(\mathcal{B}\) is a basis, \(B\) has linearly independent columns and is invertible — see matrix invertibility.
To switch between two non-standard bases \(\mathcal{B}\) and \(\mathcal{C}\):
\[ [\mathbf{v}]_\mathcal{C} = C^{-1} B \, [\mathbf{v}]_\mathcal{B}. \]
Change of basis is the algebraic machinery behind every “rotate the axes,” “decorrelate the features,” or “transform into the eigenbasis” operation.
Why Coordinates Depend on the Basis
The point in the plane labeled \((3, 1)\) does not have an intrinsic identity as “the vector \((3, 1)\)” — it has an intrinsic identity as a geometric object, and the label \((3, 1)\) is shorthand for “the coordinates of this point in the standard basis.” Choosing a different basis assigns different coordinates to the same point.
This is why the same data can look diagonal in one basis and dense in another, why principal-components analysis (writing data in the eigenbasis of the covariance) is a meaningful operation, and why the matrix of a linear map depends on the basis used to represent it.
Existence of Bases
Every finite-dimensional vector space has a basis. Two complementary procedures construct one:
- Extend an independent set. If \(\mathbf{v}_1, \ldots, \mathbf{v}_k\) are independent but do not span, pick any vector \(\mathbf{v}_{k+1}\) outside their span and add it. The enlarged set is still independent (since otherwise \(\mathbf{v}_{k+1}\) would lie in the span of the rest). Repeat until the span is all of \(V\).
- Shrink a spanning set. If \(\mathbf{v}_1, \ldots, \mathbf{v}_m\) span \(V\) but are dependent, some \(\mathbf{v}_i\) is a combination of the others; drop it. The remaining vectors still span \(V\). Repeat until independent.
Both procedures terminate after at most \(\dim V\) steps and always agree on the size of the resulting basis.