Linearly Dependent Vectors — Definition, Formula & Examples

Linearly dependent vectors are vectors where at least one can be written as a combination of the others. In other words, there is a nontrivial way to combine them using scalar multipliers so that the result is the zero vector.

A set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ in a vector space is linearly dependent if there exist scalars $c_1, c_2, \dots, c_k$ , not all zero, such that $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0}$ .

Key Formula

c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0}, \quad \text{with not all } c_i = 0

Where:

$\mathbf{v}_i$ = The vectors being tested for dependence
$c_i$ = Scalar coefficients, at least one of which is nonzero
$\mathbf{0}$ = The zero vector

How It Works

To test whether a set of vectors is linearly dependent, form a matrix with the vectors as columns and row-reduce it. If there is a column without a pivot (i.e., a free variable exists), the vectors are linearly dependent. Equivalently, if the determinant of the square matrix formed by the vectors equals zero, the vectors are linearly dependent. A quick shortcut: any set of vectors in

\mathbb{R}^n

containing more than

n

vectors is automatically linearly dependent.

Worked Example

Problem: Determine whether the vectors v₁ = (1, 2, 3), v₂ = (4, 5, 6), and v₃ = (5, 7, 9) are linearly dependent.

Step 1: Form the matrix with these vectors as columns and compute its determinant.

A = \begin{pmatrix} 1 & 4 & 5 \\ 2 & 5 & 7 \\ 3 & 6 & 9 \end{pmatrix}

Step 2: Expand the determinant along the first row.

\det(A) = 1(45-42) - 4(18-21) + 5(12-15) = 3 + 12 - 15 = 0

Step 3: Since the determinant is zero, the vectors are linearly dependent. You can verify: v₁ + v₂ − v₃ = (0, 0, 0), so c₁ = 1, c₂ = 1, c₃ = −1 is a nontrivial solution.

\mathbf{v}_1 + \mathbf{v}_2 - \mathbf{v}_3 = \mathbf{0}

Answer: The vectors are linearly dependent because det(A) = 0, and specifically v₃ = v₁ + v₂.

Why It Matters

Identifying linear dependence is central to finding bases, computing rank, and solving systems of equations in linear algebra courses. In applied settings like computer graphics and data science, dependent vectors signal redundant information, which affects dimensionality reduction techniques like PCA.

Common Mistakes

Mistake: Concluding that vectors are linearly independent just because no single vector is a scalar multiple of another.

Correction: Linear dependence can involve combinations of multiple vectors. For instance, v₃ = v₁ + v₂ makes the set dependent even though no pair is proportional. Always check the full system, not just pairwise relationships.