Bases and the Invariance of Dimension

The Intuition of Dimension

Imagine a room. To specify the location of a point inside it, you need three numbers: length, width, and height. These three directions are independent; you cannot describe "width" using only "length" and "height." Now, imagine you rotate your coordinate system so that your "x-axis" points diagonally across the floor. You still need exactly three numbers to locate the point. The room has not changed; the number of independent directions required to describe any point within it has not changed. This number is the dimension. It is an intrinsic property of the space itself, not a feature of the specific axes you chose to draw. If a space can be described by two independent directions, it is two-dimensional, no matter how you orient your rulers. This lecture formalizes this intuition by proving that all valid "rulers" (bases) for a finite-dimensional space must have the same number of "ticks" (vectors).

Defining the Dimension

A basis for a vector space \(V\) is a set of vectors that is both linearly independent and spans \(V\). This means every vector in \(V\) can be written uniquely as a linear combination of the basis vectors. The dimension of \(V\), denoted \(\dim V\), is defined as the number of vectors in any basis of \(V\). The critical question is: what if we choose a different basis? Does the number of vectors change? The Invariance of Dimension theorem answers this definitively.

Let \(B = \{ \mathbf{v}_1, \dots, \mathbf{v}_n \}\) be a basis for \(V\). Let \(B' = \{ \mathbf{w}_1, \dots, \mathbf{w}_m \}\) be another basis for \(V\). The theorem states that \(n = m\). This result is profound because it ensures that \(\dim V\) is a well-defined integer. Without this invariance, the concept of dimension would be ambiguous, and the linear algebraic structures we rely on, such as the rank of a matrix or the number of eigenvalues, would lose their meaning.

The Exchange Lemma

The proof of the Invariance of Dimension relies on a powerful tool known as the Exchange Lemma (or Steinitz Exchange Lemma). The lemma states that if \(S\) is a linearly independent set and \(T\) is a spanning set, then \(|S| \leq |T|\). In the context of bases, since a basis is both independent and spanning, we can apply the lemma in both directions.

Suppose \(B\) and \(B'\) are two bases of \(V\). Since \(B\) is independent and \(B'\) spans \(V\), the Exchange Lemma implies \(|B| \leq |B'|\). Conversely, since \(B'\) is independent and \(B\) spans \(V\), the lemma implies \(|B'| \leq |B|\). Combining these two inequalities, we must have \(|B| = |B'|\). This symmetry is the core of the invariance argument. It shows that the size of a basis is not arbitrary but is strictly constrained by the structure of the vector space.

Consequences for Linear Maps

The invariance of dimension has immediate consequences for linear maps. If \(T: V \to W\) is a linear map, the dimension of the domain and codomain dictates the possible ranks and nullities of \(T\). The Rank-Nullity Theorem states that \(\dim V = \mathrm{rank}(T) + \mathrm{nullity}(T)\). Because \(\dim V\) is invariant, the sum of the dimensions of the image and the kernel is fixed. This is crucial in physics when analyzing constraints. For example, in a system of coupled oscillators, the number of normal modes (the dimension of the configuration space) is invariant under the choice of generalized coordinates. Whether you use Cartesian coordinates or normal coordinates, the number of independent modes remains the same.

Furthermore, the concept of dimension allows us to define isomorphisms. Two vector spaces are isomorphic if and only if they have the same dimension. This means that, from a purely linear algebraic perspective, all \(n\)-dimensional vector spaces over the same field are structurally identical. This abstraction allows us to transfer results from one physical context to another, provided the dimensions match.