Linear Independence Calculator — Check if Vectors Are Linearly Independent

What Is the Linear Independence Calculator?

The linear independence calculator tells you whether a set of 2–4 vectors is linearly independent or linearly dependent. Enter the components of each vector, choose your unit context (abstract, metric, imperial, or a world currency), and the tool instantly computes the matrix rank, the determinant (for square systems), and gives you a clear verdict.

What Is a Vector?

When you ask someone "What is a vector?", the most common answer is "an arrow." After all, we usually denote vectors with an arrow over a letter: v⃗.

Formally, a vector is an element of a vector space — a set of objects that can be added together and multiplied by a scalar (a real number), obeying specific axioms. The most familiar examples are the number line, the 2D plane (pairs of numbers), and the 3D space we live in (triples of numbers).

For instance, A = (2, 3) is a 2D vector, and B = (1, −4, 7) is a 3D vector. Vector addition works component-wise: (2, 3) + (−3, 11) = (−1, 14). Scalar multiplication scales every component: 3 · (2, 3) = (6, 9).

Linear Combination of Vectors

A linear combination of vectors v₁, v₂, …, vₙ is any expression of the form:

c₁·v₁ + c₂·v₂ + … + cₙ·vₙ

where c₁, c₂, …, cₙ are real numbers (called scalars or coefficients). For example, if v₁ = (1, 0) and v₂ = (0, 1), then any 2D vector (a, b) can be written as a·v₁ + b·v₂ — a linear combination of the two basis vectors.

Linearly Independent Vectors

A set of vectors {v₁, v₂, …, vₙ} is linearly independent if the only solution to the equation

c₁·v₁ + c₂·v₂ + … + cₙ·vₙ = 0

is the trivial solution: c₁ = c₂ = … = cₙ = 0. In plain English: none of the vectors can be written as a linear combination of the others.

Conversely, vectors are linearly dependent if there exist coefficients (not all zero) such that the equation above holds — meaning at least one vector is "redundant" and can be expressed using the rest.

The Span of Vectors in Linear Algebra

The span of a set of vectors is the collection of all possible linear combinations of those vectors. If your vectors are linearly independent, their span has the same dimension as the number of vectors. If they are dependent, the span has a smaller dimension — some directions are "covered twice."

For example, vectors v₁ = (1, 2, 3), v₂ = (4, 5, 6), v₃ = (7, 8, 9) are linearly dependent (their determinant is 0), so their span is only a 2D plane inside 3D space, not all of ℝ³.

How to Check Linear Dependence

The standard method uses Gaussian elimination (row reduction):

1. Form a matrix where each vector is a row.
2. Perform row operations to reduce the matrix to row echelon form.
3. Count the number of non-zero rows — this is the rank of the matrix.
4. If rank = number of vectors → linearly independent.
   If rank < number of vectors → linearly dependent.

For square matrices (same number of vectors as dimensions), you can also compute the determinant:

• det ≠ 0 → linearly independent (the matrix is invertible).
• det = 0 → linearly dependent (the matrix is singular).

Example: Using the Linear Independence Calculator

Let's check whether these three 3D vectors are linearly independent:

• v₁ = (1, 2, 3)
• v₂ = (4, 5, 6)
• v₃ = (7, 8, 9)

Step 1 — Form the matrix:

| 1 2 3 |
| 4 5 6 |
| 7 8 9 |

Step 2 — Compute the determinant:

det = 1·(5·9 − 6·8) − 2·(4·9 − 6·7) + 3·(4·8 − 5·7)
= 1·(45 − 48) − 2·(36 − 42) + 3·(32 − 35)
= 1·(−3) − 2·(−6) + 3·(−3)
= −3 + 12 − 9 = 0

Result: det = 0, rank = 2 (not 3). The vectors are linearly dependent. Indeed, v₃ = 2·v₂ − v₁, so the third vector is a linear combination of the first two.

Now try independent vectors: v₁ = (1,0,0), v₂ = (0,1,0), v₃ = (0,0,1). Their determinant is 1 ≠ 0, rank = 3 → independent.

Unit Contexts: Metric, Imperial, and Currency

Vectors can represent physical quantities. The unit context selector adds labels to make results clearer:

• Abstract — pure mathematical vectors (x, y, z, w axes).
• Metric — coordinates in meters (e.g., force vectors in N·m, position in m).
• Imperial (US) — coordinates in feet (e.g., structural engineering in ft).
• Currency — financial portfolio vectors. Choose from 12 world currencies: USD (US Dollar), RUB (Russian Ruble), EUR, GBP, JPY, CNY, INR, BRL, CAD, AUD, CHF, KRW. Linear independence in finance means the assets in a portfolio are truly diversified (no asset can be replicated by combining the others).

FAQs

Can two vectors be linearly dependent?

Yes. Two vectors are linearly dependent if and only if one is a scalar multiple of the other (they point in the same or opposite directions). For example, (2, 4) = 2·(1, 2), so they are dependent.

What does rank tell us?

The rank of the matrix equals the dimension of the space spanned by the vectors. If rank = k < n (number of vectors), then exactly n − k vectors are "redundant."

Does linear independence depend on the order of vectors?

No. The set {v₁, v₂, v₃} is linearly independent regardless of the order you list the vectors.

What if more vectors than dimensions?

Any set of n vectors in ℝᵐ where n > m is automatically linearly dependent — you cannot have more independent directions than the space has dimensions.

Is the zero vector always dependent?

Yes. Any set that contains the zero vector is automatically linearly dependent, because you can choose the coefficient for the zero vector to be anything non-zero and still satisfy c₁·v₁ + … = 0.