Linearly Independent — Definition, Formula & Examples

Linearly independent means that no vector in a set can be written as a combination of the others. In other words, the only way to combine the vectors to get the zero vector is by using all-zero coefficients.

A set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ in a vector space is linearly independent if the equation $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0}$ has only the trivial solution $c_1 = c_2 = \cdots = c_k = 0$ . If any non-trivial solution exists, the set is linearly dependent.

Key Formula

c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0} \implies c_1 = c_2 = \cdots = c_k = 0

Where:

$\mathbf{v}_1, \dots, \mathbf{v}_k$ = Vectors in the set being tested
$c_1, \dots, c_k$ = Scalar coefficients
$\mathbf{0}$ = The zero vector

How It Works

To test whether vectors are linearly independent, place them as columns in a matrix and row-reduce to echelon form. If every column contains a pivot (leading 1), the vectors are linearly independent. Equivalently, the matrix has a nonzero determinant when it is square. If any column lacks a pivot, a free variable exists, meaning a non-trivial linear combination equals the zero vector, so the set is linearly dependent.

Worked Example

Problem: Determine whether the vectors

\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}

\mathbf{v}_2 = \begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix}

, and

\mathbf{v}_3 = \begin{pmatrix} 1 \\ 1 \\ 5 \end{pmatrix}

are linearly independent.

Step 1: Form a matrix with these vectors as columns and set up the augmented system for the equation

c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + c_3\mathbf{v}_3 = \mathbf{0}

\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 2 & 3 & 5 \end{pmatrix}

Step 2: Row-reduce. Replace

R_3

with

R_3 - 2R_1

, then replace the new

R_3

with

R_3 - 3R_2

\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix}

Step 3: The third column has no pivot, so

c_3

is a free variable. Setting

c_3 = 1

gives

c_1 = -1

and

c_2 = -1

, a non-trivial solution.

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

Answer: The vectors are linearly dependent because

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

Why It Matters

Linear independence determines whether a set of vectors forms a basis for a vector space, which is central to solving systems of equations, computing matrix inverses, and understanding dimension. In applied fields like data science and engineering, checking independence reveals whether measurements or signals carry redundant information.

Common Mistakes

Mistake: Assuming vectors are linearly independent just because none of them are scalar multiples of each other.

Correction: Two vectors not being multiples only rules out pairwise dependence. A third vector could still equal a combination of the first two, as in the example above. Always solve the full system or compute the determinant.