Linear Dependence and Independence

linear independence

Definition

Linear combination

Let \(S = \{v_1, \cdots, v_n\}\) be a set of vectors in \(V\). If \(a_1, \cdots, a_n\) are scalars, then the following expression is called a linear combination: \[ a_1 v_1 + \cdots + a_n v_n \] Linear dependence

Let \(S = \{v_1, \cdots, v_n\}\) be a set of vectors in \(V\). \(S\) is said to be linearly dependent if we can find scalars \(a_1, \cdots, a_n\), with at least one \(a_i \neq 0\) such that: \[ a_1 v_1 + \cdots + a_nv_n = 0 \] Linear independence

Let \(S = \{v_1, \cdots, v_n\}\) be a set of vectors in \(V\). \(S\) is said to be linearly independent if for every set of scalars \(a_1, \cdots, a_n\): \[ a_1v_1 + \cdots + a_n v_n = 0 \implies a_1 = \cdots = a_n = 0 \]

Examples

  • \(\{(1, 2), (2, 4)\}\) is linearly dependent in \(\mathbb{R}^{2}\).
  • \(\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}\) is linearly independent in \(\mathbb{R}^{3}\).

Useful results

  • If \(0 \in S\) then \(S\) is linearly dependent.
  • If \(S = \{u\}\) with \(u \neq 0\), then \(S\) is linearly independent.
  • If \(S = \{u, cu\}\) is linearly dependent where \(c \in \mathbb{R}\).
  • If \(S\) is linearly dependent, then every superset of \(S\) is linearly dependent.
  • If \(S\) is linearly independent, then every subset of \(S\) is linearly independent.

Algorithm

Given a set of vectors, one way to determine linear (in)dependence of a set \(S\) is as follows:

  • Add each vector of \(S\) as the row of a matrix. Call the matrix \(A\).
  • Obtain the RREF of \(A\). Call this matrix \(R\).
  • If \(R\) has no zero rows, then \(S\) is linearly independent. If \(R\) has a zero row, then \(S\) is linearly dependent.