BYTETOOLS

Linear Independence Calculator

Test whether a set of vectors is linearly independent by comparing rank to vector count, and see the explicit relation when they turn out to be dependent.

Dependent
Verdict
2
Rank
3
Vectors
3
Dimension of each

Linearly dependent

The 3 vectors row-reduce to only 2 pivots, which is fewer than the 3 vectors — so at least one vector is a combination of the others. There are 1 independent relation among them.

Explicit dependency relations

v1 − 2·v2 + v3 = 0(v3 is redundant — it is spanned by the others)

Each relation comes from a free variable in the system c₁v₁ + … = 0. Rearranging any relation writes one vector in terms of the rest.

Row reduction of the stacked vectors

10-1
012
000

Each input vector is one row. Rank 2 = the number of non-zero rows left after Gauss-Jordan elimination, computed in exact fractions.

What is the Linear Independence Calculator?

The ByteTools Linear Independence Calculator takes a list of vectors, stacks them into a matrix, row-reduces it in exact fractions and compares the rank to the number of vectors you entered.

  • Yes or no verdict backed by the rank compared against the vector count
  • Explicit dependency relations like v₁ − 2v₂ + v₃ = 0, not just a verdict
  • Names which vector is redundant for each relation
  • Flags the automatic case of more vectors than dimensions
  • Exact fraction arithmetic, so near-zero rounding never fakes independence
  • Everything computed locally in your browser

How to use the Linear Independence Calculator

  1. 1

    Enter one vector per line, with components separated by spaces or commas.

  2. 2

    Read the verdict tile — Independent or Dependent — with the rank next to it.

  3. 3

    If the set is dependent, read the explicit relations showing which vector is redundant.

  4. 4

    Check the row reduction panel to see the rank being counted.

  5. 5

    Copy the relations with the copy button for your working.

About the Linear Independence Calculator

The ByteTools Linear Independence Calculator takes a list of vectors, stacks them into a matrix, row-reduces it in exact fractions and compares the rank to the number of vectors you entered. If they match, the vectors are independent — no one of them can be built from the others. If the rank is smaller, they are dependent, and the tool goes further than a yes/no answer.

For dependent sets it solves c₁v₁ + c₂v₂ + … = 0 and prints each independent relation explicitly, for example v₁ − 2v₂ + v₃ = 0, naming which vector is redundant. That turns an abstract verdict into something you can use in a proof or a homework write-up. It also points out the shortcut case where you simply have more vectors than dimensions, which forces dependence no matter what the values are.

The whole computation is JavaScript running in your browser. Your vectors are never uploaded, no account is needed, and the page works offline.

Frequently asked questions

How do you tell if vectors are linearly independent?

Put the vectors into a matrix and row-reduce it. If the rank equals the number of vectors, they are independent; if the rank is smaller, at least one vector is a combination of the others. This calculator performs that reduction and shows you the result.

Can 3 vectors in R2 be linearly independent?

No. You can never have more than n independent vectors in n-dimensional space, so three vectors in the plane are always dependent. The tool calls this out directly whenever you enter more vectors than components.

What does linearly dependent actually mean?

It means there is some combination of the vectors, with coefficients that are not all zero, that adds up to the zero vector. Equivalently, at least one vector lies in the span of the others and adds no new direction to the set.

Is the zero vector linearly independent?

Never. Any set containing the zero vector is automatically dependent, because 1 times the zero vector already equals zero — a non-trivial relation. A single non-zero vector, by contrast, is always independent on its own.

Does linear independence mean the vectors are perpendicular?

No — perpendicular vectors are always independent, but independent vectors need not be perpendicular. (1,0) and (1,1) are independent and clearly not at right angles. If you want a perpendicular set, run them through the Gram-Schmidt calculator.

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