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

\[ \mathbf{w} = k_1 \mathbf{v}_1 + k_2 \mathbf{v}_2 + \ldots + k_n \mathbf{v}_n \]

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.

Example 1

Determine whether the following sets of vectors are linearly dependent:

\( \mathbf{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix} \), \( \mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix} \), and \( \mathbf{c} = \begin{pmatrix} 5 \\ 6 \end{pmatrix} \)

Solution:

Note that vectors \( \mathbf{a} \) and \( \mathbf{b} \) are not parallel.

Suppose vector \( \mathbf{c} \) can be expressed as a linear combination of vectors \( \mathbf{a} \) and \( \mathbf{b} \), i.e., \( \mathbf{c} = m \mathbf{a} + n \mathbf{b} \).

Then, we have the following system of equations:

\[ 5 = 2m + 3n \]

\[ 6 = m - n \]

Solving these simultaneous equations, we find:

\[ m = \frac{23}{5} \]

\[ n = -\frac{7}{5} \]

Since \( m \) and \( n \) are not both zero, the set of vectors is linearly dependent.

Note: In general, any set of three or more two-dimensional vectors is linearly dependent.

Example 2

Determine whether the following sets of vectors are linearly dependent:

a = 

 [ 3 ]
     [ 4 ]
     [-1]

b = 

 [ 2 ]
     [ 1 ]
     [ 3 ]

c = 

 [-1]
     [ 0 ]
     [ 1 ]

Solution:

Note that \( \mathbf{a} \) and \( \mathbf{b} \) are not parallel.

Suppose \( \mathbf{c} = m\mathbf{a} + n\mathbf{b} \)

Then \( -1 = 3m + 2n \)

0 = 4m + n

1 = -m + 3n

Solving the first two equations, we have

\( m = \frac{1}{5} \) and \( n = -\frac{4}{5} \)

But these values do not satisfy the third equation, as \( -m + 3n = -\frac{13}{5} \neq 1 \).

The three equations have no solution, so the vectors are linearly independent.

Example 3

Points A and B have position vectors a and b respectively, relative to an origin O.

The point D is such that OD = kOA and the point E is such that AE = `AB. The line segments BD and OE intersect at X.

Assume that OX = (2/5)OE and XB = (4/5)DB.


Express \( \overrightarrow{XB} \) in terms of \( \vec{a} \), \( \vec{b} \), and \( k \).

Solution

\( \overrightarrow{XB} = \frac{4}{5} \overrightarrow{DB} = \frac{4}{5} (\overrightarrow{OD} + \overrightarrow{OB}) = \frac{4}{5} (-k\overrightarrow{OA} + \overrightarrow{OB}) = \frac{4}{5}(-ka + b) = -\frac{4k}{5}\vec{a} + \frac{4}{5}\vec{b} \)

Express \( \overrightarrow{OX} \) in terms of \( \vec{a} \), \( \vec{b} \), and \( l \).

Solution

\( \overrightarrow{OX} = \frac{2}{5} \overrightarrow{OE} = \frac{2}{5} (\overrightarrow{OA} + \overrightarrow{AE}) = \frac{2}{5} \overrightarrow{OA} + \frac{2}{5} \overrightarrow{AB} = \frac{2}{5}a + \frac{2}{5}\cdot(`b - a) \)

Example 4

From previous example:

Express \( \overrightarrow{XB} \) in terms of \( \vec{a} \), \( \vec{b} \), and \( l \).

Solution

\( \overrightarrow{cXB} = \overrightarrow{XO} + \overrightarrow{OB} = -\overrightarrow{OX} + \overrightarrow{OB} = -\frac{2}{5}(1 - `)a - \frac{2}{5}b + b = \frac{2}{5}(` - 1)a + (1 - \frac{2}{5})b \)

Find \( \vec{k} \), and \( l \).

Solution:

As \( \mathbf{a} \) and \( \mathbf{b} \) are linearly independent vectors, the vector \( \overrightarrow{XB} \) has a unique representation in terms of \( \mathbf{a} \) and \( \mathbf{b} \). From parts a and c, we have

\( -\frac{4k}{5}a + \frac{4}{5}b = \frac{2}{5}(` - 1)a + \left(1 - \frac{2}{5}\right)b \)

Hence

\( -\frac{4k}{5} = \frac{2}{5}(` - 1) \) (1) and \( \frac{4}{5} = 1 - \frac{2}{5} \) (2)

From equation (2), we have

\( \frac{2}{5}` = \frac{1}{5} \Rightarrow ` = \frac{1}{2} \)

Substitute in (1):

\( -\frac{4k}{5} = \frac{2}{5}\left(\frac{1}{2} - 1\right) \Rightarrow k = \frac{1}{4} \)

Exercise &&1&& (&&1&& Question)

For a set of two vectors \( \mathbf{a} \) and \( \mathbf{b} \), under what condition are they considered linearly dependent?

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Exercise &&2&& (&&1&& Question)

Consider three vectors in two-dimensional space: \( \mathbf{v}_1 = \langle 1, 2 \rangle \), \( \mathbf{v}_2 = \langle -2, 4 \rangle \), and \( \mathbf{v}_3 = \langle 3, -6 \rangle \). Are these vectors linearly dependent or linearly independent?

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