AOS5 Topic 3: Linear Dependence and Independence

A vector \( \mathbf{w} \) is a linear combination of vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \) if it can be expressed in the form

for some real numbers \( k_1, k_2, \ldots, k_n \).

Linearly dependent

A set of vectors is said to be linearly dependent if at least one of its members can be expressed as a linear combination of other vectors in the set.

Linearly independent

A set of vectors is said to be linearly independent if it is not linearly dependent.That is, a set of vectors is linearly independent if no vector in the set is expressible as a linear combination of other vectors in the set.

For example, it is easy to show that a set of two non-zero vectors is linearly dependent if andonly if the two vectors are parallel.

We can give a useful alternative description of linear dependence:

Two vectors: A set of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is linearly dependent if and only if there exist real numbers \( k \) and \( \ell \), not both zero, such that \( k\mathbf{a} + \ell\mathbf{b} = \mathbf{0} \).

Three vectors: A set of three vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) is linearly dependent if and only if there exist real numbers \( k \), \( \ell \), and \( m \), not all zero, such that \( k\mathbf{a} + \ell\mathbf{b} + m\mathbf{c} = \mathbf{0} \).

In general: A set of \( n \) vectors \( \mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n \) is linearly dependent if and only if there exist real numbers \( k_1, k_2, \ldots, k_n \), not all zero, such that \( k_1\mathbf{a}_1 + k_2\mathbf{a}_2 + \ldots + k_n\mathbf{a}_n = \mathbf{0} \).

Note: Any set that contains the zero vector is linearly dependent. Any set of three or more two-dimensional vectors is linearly dependent. Any set of four or more three-dimensional vectors is linearly dependent.

We will use the following method for checking whether three vectors are linearly dependent.

Linear dependence for three vectors:

Let \( \mathbf{a} \) and \( \mathbf{b} \) be non-zero vectors that are not parallel. Then vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) are linearly dependent if and only if there exist real numbers \( m \) and \( n \) such that \( \mathbf{c} = m\mathbf{a} + n\mathbf{b} \).

This representation of a vector \( \mathbf{c} \) in terms of two linearly independent vectors \( \mathbf{a} \) and \( \mathbf{b} \) is unique, as demonstrated in the following important result.

Linear combinations of independent vectors:

Let \( \mathbf{a} \) and \( \mathbf{b} \) be two linearly independent (i.e., not parallel) vectors. Then:

\( m\mathbf{a} + n\mathbf{b} = p\mathbf{a} + q\mathbf{b} \) implies \( m = p \) and \( n = q \)

Proof:

Assume that \( ma + nb = pa + qb \). Then \( (m - p)a + (n - q)b = 0 \). As vectors \( \mathbf{a} \) and \( \mathbf{b} \) are linearly independent, it follows from the definition of linear independence that \( m - p = 0 \) and \( n - q = 0 \). Hence \( m = p \) and \( n = q \).

Note: This result can be extended to any finite number of linearly independent vectors.