Linear Algebra
2.1110 min read

The Hammer Lemma and Span Structure

A key lemma connects dependence to span: if v\mathbf{v} appears with a nonzero coefficient in a dependence relation among {a1,…,ak,v}\{\mathbf{a}_1, \ldots, \mathbf{a}_k, \mathbf{v}\}, then v∈Span({a1,…,ak})\mathbf{v} \in \text{Span}(\{\mathbf{a}_1, \ldots, \mathbf{a}_k\}).

This means: in a linearly dependent set, at least one vector is redundant — it can be removed without shrinking the span. This is the pull-off lemma: you can peel away one vector from a dependent set and preserve the span.

Conversely, if v∉Span(S)\mathbf{v} \notin \text{Span}(S), then adding v\mathbf{v} to SS while preserving independence is always possible (push-on lemma). These two lemmas are the combinatorial engine behind all basis theory.

Formal View

Lemma 2.11 (Hammer) — Dependence Implies Membership in Span
If c0v+c1a1+⋯+ckak=0c_0\mathbf{v} + c_1\mathbf{a}_1 + \cdots + c_k\mathbf{a}_k = \mathbf{0} with c0≠0c_0 \neq 0, then
v=−c1c0a1−⋯−ckc0ak∈Span({a1,…,ak})\mathbf{v} = -\frac{c_1}{c_0}\mathbf{a}_1 - \cdots - \frac{c_k}{c_0}\mathbf{a}_k \in \text{Span}(\{\mathbf{a}_1, \ldots, \mathbf{a}_k\})
Corollary 2.11 (Pull-off)
If {a1,…,ak}\{\mathbf{a}_1, \ldots, \mathbf{a}_k\} is linearly dependent, then some ai\mathbf{a}_i lies in the span of the others, and Span({a1,…,ak})=Span({a1,…,ak}∖{ai})\text{Span}(\{\mathbf{a}_1, \ldots, \mathbf{a}_k\}) = \text{Span}(\{\mathbf{a}_1, \ldots, \mathbf{a}_k\} \setminus \{\mathbf{a}_i\}).

Why This Matters

The Hammer Lemma is the key tool for proving basis-related theorems and the Rank-Nullity theorem.

  • Data compression: removing linearly dependent features without losing information
  • Signal processing: identifying and removing redundant sensors
  • Dimensionality reduction: PCA exploits the pull-off principle to find minimal spanning sets

Quiz

Question 1

If {a,b,c}\{\mathbf{a}, \mathbf{b}, \mathbf{c}\} is linearly dependent, then removing one vector always reduces the span.

Common Mistakes

  • Thinking all vectors in a dependent set are redundant — only at least one is guaranteed to be removable.